Introduction

Let H 0 : C → C denote the function

where ξ denotes the Riemann xi function

and ζ is the Riemann zeta function. Then H 0 is an entire even function with functional equation H 0 (z) = H 0 (z), and the Riemann hypothesis is equivalent to the assertion that all the zeros of H 0 are real. It is a classical fact (see [29, page 255]) that H 0 has the Fourier representation

where Φ is the super-exponentially decaying function

The sum defining Φ(u) converges absolutely for negative u also. From Poisson summation, one can verify that Φ satisfies the functional equation Φ(u) = Φ(-u) (that is, Φ is even).

De Bruijn [4] introduced the more general family of functions H t : C → C for t ∈ R by the formula

As noted in [11, page 114], one can view H t as the evolution of H 0 under the backward heat equation ∂ t H t (z) = -∂ zz H t (z). As with H 0 , each of the H t is an entire even function with functional equation H t (z) = H t (z). From results of Pólya [20], it is known that if H t has purely real zeros for some t, then H t has purely real zeros for all t > t. De Bruijn showed that the zeros of H t are purely real for t 1/2. Strengthening these results, Newman [17] showed that there is an absolute constant -∞ < Λ 1/2, now known as the De Bruijn-Newman constant, with the property that H t has purely real zeros if and only if t Λ. The Riemann hypothesis is then clearly equivalent to the upper bound Λ 0. Newman conjectured the complementary lower bound Λ 0 and noted that this terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fmp.2020.6 Downloaded from https://www.cambridge.org/core. IP address: 107.190.7.212, on 25 Aug 2020 at 14:47:54, subject to the Cambridge Core

The De Bruijn-Newman constant is non-negative conjecture asserts that if the Riemann hypothesis is true, it is only 'barely so'. As progress towards this conjecture, several lower bounds on Λ were established; see Table 1.

Table 1. Previous lower bounds on Λ. Dates listed are publication dates. The final four results use the method of Csordas, Smith and Varga [11]. Csordas-Smith-Varga 1994 [11] -5.895 × 10 - 9 Csordas-Odlyzko-Smith-Varga 1993 [9] -2.63 × 10 - 9 Odlyzko 2000 [19] -1.15 × 10 - 11 Saouter-Gourdon-Demichel 2011 [23] We also mention that the upper bound Λ 1/2 of de Bruijn [4] was sharpened slightly by Ki, Kim and Lee [13] to Λ < 1/2. (In press: this bound has recently been improved to Λ 0.22 in [21].) See also [5,25] on work on variants of Newman's conjecture and [3,Ch. 5] for a survey.

The main result of this paper is to affirmatively settle Newman's conjecture: THEOREM 1. One has Λ 0.

We now discuss the methods of the proof. Starting from the work of Csordas-Smith-Varga [11], the best lower bounds on Λ were obtained by exploiting the following repulsion phenomenon: if Λ was significantly less than zero, then adjacent zeros of H 0 (or of the Riemann ξ function) cannot be too close to each other (as compared with the other nearby zeros). See [11, Theorem 1] for a precise statement. In particular, a negative value of Λ gives limitations on the quality of 'Lehmer pairs' [14], which roughly speaking refer to pairs of adjacent zeros of the Riemann zeta function that are significantly closer to each other than the average spacing of zeros at that level. The lower bounds on Λ in [9,11,19,23] then follow from numerically locating Lehmer pairs of increasingly high quality. (See also [26] for a refinement of the Lehmer pair concept used in the above papers.)

In principle, one could settle Newman's conjecture by producing an infinite sequence of Lehmer pairs of arbitrarily high quality. As suggested in [19], we were able to achieve this under the Gaussian Unitary Ensemble (GUE) hypothesis on the asymptotic distribution of zeros of the Riemann zeta function; we do not detail this computation here as it is superseded by our main result. (A sketch of the argument may be found at terrytao.wordpress.com/2018/01/20.) However, without the GUE hypothesis, the known upper bounds on narrow gaps between zeros (for example, [7]) do not appear to be sufficient to make this strategy work, even if one assumes the Riemann hypothesis (which one can do for Theorem 1 without loss of generality). Instead, we return to the analysis in [11] and strengthen the repulsion phenomenon to a relaxation to local equilibrium phenomenon: if Λ is negative, then the zeros of H 0 are not only repelled from each other but will nearly always be arranged locally as an approximate arithmetic progression, with the gaps between zero mostly staying very close to the global average gap that is given by the Riemann-von Mangoldt formula. (To illustrate the equilibrium nature of arithmetic progressions under backward heat flow, consider the entire functions F t (z) := e tu 2 cos(zu) for some fixed real u > 0. These functions all have zeros on the arithmetic progression { To obtain the local relaxation to equilibrium under the hypothesis that Λ < 0 requires a sequence of steps in which we obtain increasingly strong control on the distribution of zeros of H t for Λ < t 0 (actually for technical reasons, we will need to move t away from Λ as the argument progresses, restricting instead to ranges such as Λ/2 t 0 or Λ/4 t 0). The first step is to obtain Riemann-von Mangoldt type formulae for the number of zeros of H t in an interval such as [0, T ] or [T, T + α], where T 2 and 0 < α o(T ). When t = 0, we can obtain asymptotics of T 4π log T 4π -T 4π + O(log T ) and α 4π log T +o(log T ) by the classical Riemann-von Mangoldt formula and a result of Littlewood, respectively; this gives good control on the zeros down to length scales α 1. For Λ < t < 0, we were only able to obtain the weaker bounds of T 4π log T 4π -T 4π + O(log 2 T ) and α 4π log T + o(log 2 T ), respectively, down to length scales α log T , but it turns out that these bounds still (barely) suffice for our arguments; see Section 3. A key input in the proof of the Riemann-von Mangoldt type formula will be some upper and lower bounds for H t (x -i y) when y is comparable to log x; see Lemma 4 for a precise statement. The main tool used to prove these bounds is the saddle point method, in which various contour integrals are shifted until they resemble the integral for the gamma function, to which the Stirling approximation may be applied.

It was shown in [11] that in the region Λ < t 0, the zeros x j (t) of H t are simple, and furthermore evolve according to the system of ordinary differential terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fmp.2020.6 Downloaded from https://www.cambridge.org/core. IP address: 107.190.7.212, on 25 Aug 2020 at 14:47:54, subject to the Cambridge Core

The De Bruijn-Newman constant is non-negative equations (ODEs)

see Theorem 11 for a more precise statement. One can view this equation as describing the dynamics of a system of 'particles' x j , in which every pair of particles x j , x k experiences a repulsion that is inversely proportional to their separation. (We caution however that the dynamics here are not Newtonian in nature since (5) prescribes the velocity ∂ t x k of each particle rather than the acceleration ∂ 2 t x k . Nevertheless we found the physical analogy to be helpful in locating the arguments used in this paper.) By refining the analysis in [11], we can obtain a more quantitative lower bound on the gap x j+1 (t) -x j (t) between adjacent 'particles' (zeros), in particular establishing a bound of the form log 1 x j+1 (t) -x j (t) log 2 j log log j for all large j in the range Λ/2 t 0; see Proposition 13 for a more precise statement. While far from optimal, this bound almost allows one to define the Hamiltonian H(t) := j,k: j =k log 1

, although in practice we will have to apply some spatial cutoffs in j, k to make this series absolutely convergent. For the sake of this informal overview, we ignore this cutoff issue for now. The significance of this quantity is that system (5) can (formally, at least) be viewed as the gradient flow for the Hamiltonian H(t). In particular, there is a formal monotonicity formula

where the energy E(t) is defined as

Again, in practice, one needs to apply spatial cutoffs to j, k to make this quantity finite, and one then has to treat various error terms arising from this cutoff, which among other things 'renormalizes' the summands 1 |x j (t)-x k (t)| 2 so that the renormalized energy vanishes when the zeros are arranged in the equilibrium state of an arithmetic progression; we ignore these issues for the current discussion. A further formal calculation indicates that E(t) is monotone nonincreasing in time (so that H(t) is formally convex in time, as one would expect for the gradient flow of a convex Hamiltonian). Exploiting (a variant terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fmp.2020.6 Downloaded from https://www.cambridge.org/core. IP address: 107.190.7.212, on 25 Aug 2020 at 14:47:54, subject to the Cambridge Core of) equation ( 6), we are able to control integrated energies that resemble the quantities 0 Λ/2 E(t) dt; see first the weak preliminary integrated energy bound in Proposition 15 and then the final integrated energy bound in Theorem 17. By exploiting local monotonicity properties of the energy (and using a pigeonholing argument of Bourgain [2]), we can then obtain good control (a truncated version) of the energy E(t) at time t = 0, which intuitively reflects the assertion that the 'particles' x j (t) are close to local equilibrium at time t = 0. This implies that the zeros of the Riemann zeta function behave locally like an arithmetic progression on the average. However, this can be ruled out by the existing results on the local distribution of zeros, such as pair correlation estimates of Montgomery [15]. As it turns out, it will be convenient to make use of a closely related estimate of Conrey, Ghosh, Goldston, Gonek and Heath-Brown [6].

It may be possible to use the methods of this paper to also address the generalized Newman conjecture introduced in [25], but we do not pursue this direction here. REMARK 2. It is interesting to compare this with the results in [13, Theorem 1.14], which show that regardless of the value of Λ, the zeros of H t will be spaced like an arithmetic progression on average for any positive t. REMARK 3. In press: We note that in forthcoming work, Alex Dobner has found a proof that Λ 0, which avoids the heat equation approach we have used here. Dobner's approach instead relies on a Riemann-Siegel type approximation for H t in order to demonstrate the existence of zeros off the critical line. There is also some very intriguing numerical work of Rudolph Dwars (see the comments to terrytao.wordpress.com/2018/12/28) that suggests that many of the zeros of H t , t < 0 away from the critical line organize around deterministic curves.

1.1. Notation. Throughout the rest of the paper, we will assume for the sake of contradiction that Newman's conjecture fails:

In particular, this implies the Riemann hypothesis (which, as mentioned previously, is equivalent to the assertion Λ 0).

We will have a number of logarithmic factors appearing in our upper bounds. for several of these bounds. We also use the standard branch of the complex logarithm, with imaginary part in the interval (-π, π], and the standard branch z 1/2 := exp( 1 2 log z) of the square root, defined using the standard branch of the complex logarithm.

Let Λ < t 0; then the zeros of H t are all real and symmetric around the origin. It is a result of Csordas, Smith and Varga [11,Corollary 1] that the zeros are also distinct and avoid the origin. Thus we can express the zeros of H t as (x j (t)) j∈Z * , where Z * := Z\{0} are the nonzero integers, 0 < x 1 (t) < x 2 (t) < . . . , and x -j (t) = -x j (t) for all j 1. For any real numbers j -j + , we use [ j -, j + ] Z * to denote the discrete interval

We For X and Y depending on an asymptotic parameter T , we will also use the notation X Y or X = Õ(Y ) for X Y log O (1) T in the last two sections of this paper.

Furthermore, in sums that will appear which depend on a parameter T , we say that indices j, k are nearby, and write j ∼ T k if one has 0 < | j -k| < (T 2 + | j| + |k|) 0.1 . We will use a marked sum to indicate principle value summation:

In cases where there is any chance of confusion for the range of summation, we record the index being summed and use a colon to indicate its range; for example, we write j: j =k to indicate that the summation is over j, and j is to not equal k (where k is fixed outside the sum). Semicolons are used to separate additional conditions. We use the phrase for almost every t throughout this paper to denote that a relation holds for all t except a set of null Lebesgue measure.

Asymptotics of H t

In this section, we establish some upper and lower bounds on H t (z) and its logarithmic derivative H t H t (z). We will be able to obtain reasonable upper bounds in the regime where z = xi y with y = O(log + x), and obtain more precise asymptotics when y log + x (as long as the ratio y/ log + x is large enough); this will be the key input for the Riemann-von Mangoldt type asymptotics in the next section. More precisely, we show the following. LEMMA 4. Let z = xiκ log + x for some x 0 and 0 κ C, and let Λ < t 0. Then one has

(The reader is advised not to take the numerous factors of π , √ 2 and so on appearing in this section too seriously, as the exact numerical values of these constants are not of major significance in the rest of the arguments.) Furthermore, there is an absolute constant C > 0 (not depending on C) such that if κ C , then one has the refinement

as well as the additional estimate

using the standard branch of the complex logarithm.

REMARK 5. With a little more effort, one could replace the hypothesis Λ < t here by -C < t; in particular (in contrast to the remaining arguments in this paper), these results are nonvacuous when Λ 0. However, we will need to assume Λ < t in the application of these estimates in the next section, particularly with regard to the proof of (49). Our proof methods also allow for a more precise version of asymptotic (8) (as one might expect given the level of precision in (9)), but such improvements do not seem to be helpful for the rest of the arguments in this paper. In the t = 0 case, one can essentially obtain Dirichlet series expansions for 1 H 0 (z) or H 0 H 0 (z), which allow one to also obtain bounds such as (8) or (9) when the imaginary part of z is much smaller than log + x. However, in the t < 0 case, there does not appear to be any usable series expansions for 1 H t (z) or H t H t (z) that could be used to prove (8) or (9). Instead, we will prove these estimates by computing H t (z) to a high degree of accuracy, which we can only The De Bruijn-Newman constant is non-negative do when y is greater than or equal to a large multiple of log + x in order to ensure that the series expansions we have for H t (z) converge rapidly.

We begin by treating the easy case t = 0, in which we can exploit identity (1). We have the very crude bound

whenever σ 1/2 and τ ∈ R (this follows for instance from [29,Theorem 4.11]). In the region σ 1/4, we also have the Stirling approximation (see for example, [1, 6.1.41])

where we use the standard branch of the logarithm; in particular,

As arctan τ σ = π 2 sgn(τ ) + O( σ σ +|τ | ), we have in particular that

. Inserting these bounds into (1) and (2), we obtain the crude upper bound

for x ∈ R and y 0. This gives the s = 0 case of (7). As is well known, when σ 2 (say), we can improve (10) to

and so we obtain the improvement

when y C log + x (in fact, in this case it would suffice to have y 4, say). This gives the s = 0 case of (8). Finally, from taking logarithmic derivatives of (1) and ( 2), one has Henceforth, we address the t < 0 case. We begin with the proof of the upper bound (7). Here it will be convenient to exploit the fundamental solution for the (backward) heat equation to relate H t with H 0 . Indeed, for any t < 0, we have the classical heat equation (or Gaussian) identity

for any complex numbers z, u; replacing z, r by -z, -r and averaging, we conclude that e tu 2 cos(zu) = 1 √ 4π R e -r 2 /4 cos (z + r |t| 1/2 )u dr.

Multiplying by Φ(u), integrating u from 0 to infinity and using Fubini's theorem, we conclude that

Applying (13), the triangle inequality and the hypothesis Λ < t 0, we conclude that

Using Since y = O C (log + x), this gives (7). To prove the remaining two bounds (8) and (9), it is convenient to cancel off the t = 0 case that has already been established, and reduce to showing that

and

when κ C . To prove these estimates, the heat equation approach is less effective due to the significant oscillation present in H 0 . Instead, we will use the method of steepest descent (also known as the saddle point method) to shift contours to where the phase is stationary rather than oscillating. We allow all implied constants to depend on C. We may assume that x is larger than any specified constant C (depending on C) as the case x = O C (1) follows trivially from compactness, since the zeros of H t for t Λ are all real, so that H t (z) is bounded away from zero in this region of interest. Now suppose that z = xi y, where y = κ log + x for some C κ C; in particular, C is large since C is. As Φ is even, we may write (4) as

From (3) and Fubini's theorem (which can be justified when t < 0), we conclude that

where I t (b, ζ ) denotes the oscillatory integral

which is an absolutely convergent integral for t < 0 whenever Re b > 0.

We therefore need to obtain good asymptotics on I t (b, ζ ) for b 1 and ζ in the region

Observe that the phase tw 2 -be 4w + ζ w has a stationary point at the origin when 4b = ζ . In general, 4b will not equal ζ ; however, for any complex number w 0 in the strip

we see from shifting the contour in (19) to the horizontal line {w + w 0 : w ∈ R} that we have the identity

whenever b > 0 (so that be 4w 0 has a positive real part). We will thus be able to reduce to the stationary phase case 4b = ζ if we can solve the equation

in strip (21). This we do in the following lemma. (One could also write w 0 explicitly in terms of the Lambert W -function as

), but we will not use this expression in this paper, and in fact will not explicitly invoke any properties of the W -function in our arguments.) LEMMA 6. If b 1 and ζ ∈ Ω, then there exists a unique w 0 = w 0 (b, ζ ) in strip (21) such that (23) holds. Furthermore, we have the following estimates: x exp(100 x 1/2 |t| ), then

where the superscript in the O() notation indicates that these quantities are real-valued. Proof. The function w 0 → 4be 4w 0 -2tw 0 traverses the graph {a + i( π |t| 4 + 4be 2a/|t| ) : a ∈ R} on the upper edge { a 2|t| + i π 8 : a ∈ R} of strip (21), while the lower edge of the strip is of course mapped to the real axis. Since |t| Λ and C, C are large, the region Ω lies between these two curves, and so from the argument principle (and observing that the map w 0 → 4be 4w 0 -2tw 0 sends the line segments {-R + iβ : 0 < β < π/8} and {R + iβ : 0 < β < π/8} well to the left and right of ζ , respectively, for R large enough), for every ζ ∈ Ω, there exists exactly one w 0 in strip (21) such that 4be 4w 0 -2tw 0 = ζ , which is of course equivalent to (23). The uniqueness implies that the holomorphic function w 0 → 4be 4w 0 -2tw 0 has a nonzero derivative at this value of w 0 . Now write ζ = y + i x as per (20), and write w 0 = α + iβ for some α ∈ R and 0 < β < π/8. Taking real and imaginary parts in (23), we have the system of equations 4be 4α cos 4β = y + 2tα ( 24)

To prove (i), suppose for contradiction that Re(4be 4w 0 ) < 1; thus

Since t, β = O(1), we see from ( 25) that 4be 4α sin 4β

x, and hence from sin 2 4β + cos 2 4β = 1, we have 4be 4α x and hence (since b 1) α 1 4 log + x + O(1). In particular, -2tα |t| 2 log + x + O(1). Inserting this into (24) and using (26), one then has

which contradicts (20) since |t| Λ and C is large. Now we show (ii). From ( 25) and sin 4β 1, t,

and hence on taking logarithms (and using the fact that b 1 and x is large)

On the other hand, from squaring ( 24) and ( 25) and summing, we have

Crudely bounding x +2tβ = O(x), y = O(x), b 1 and t = O(1), we conclude that e 8α

x 2 + α 2 , which implies that α O(log + x). From the hypothesis b x exp(100 x 1/2 |t| ) and ( 27), we also have α -O(x 1/2 /t); thus tα x 1/2 . Returning to (28) and using 2tβ = O(1) and y

x 1/2 , we conclude that

So on taking square roots

and hence on taking logarithms, we have the matching upper bound

to (27). In particular,

Inserting this and ( 29) into ( 24), we have

and hence (by Taylor expansion of the arc cosine function)

Finally, we prove (iii). From identity (28) and crudely bounding y, tβ = O(x), we have

Under the hypothesis b > x exp( x 1/2 |t| ), so that 1/|t| and x are O(b 1/10 ) (say), so both options force -α 1 8 log b as claimed.

We combine the above lemma with the following asymptotic.

LEMMA 7. Let b be a complex number with Re b 1. Then

using the standard branch of the square root.

Proof. One could establish this from Laplace's method, but we will instead use the Stirling approximation (11). (We thank Alex Dobner for pointing out some issues in the original proof of this lemma and suggesting a repaired proof, which is reproduced here.) Writing

, where µ is the Gaussian probability measure

of mean zero and variance |t|/8, and applying Fubini's theorem, we obtain

Making the change of variables r = be 4w (and contour shifting or analytic continuation) and the definition

and hence

We divide the integral into regions |ξ | 10|t| 1/2 |b| 1/2 and |ξ | > 10|t| 1/2 |b| 1/2 . By applying the Stirling approximation, the integral over the first region becomes

Note that we may assume that |b| is sufficiently large so that |ξ | < |b|/2 in this region (the small |b| case of the lemma follows trivially from compactness), and so we have 1

|b| . Substituting these expressions into the integrand, we get

and now the integral evaluates to 1 + O( 1 |b| ). Thus it will suffice to establish the tail bound

By applying the triangle inequality and bounding the integrand with

(say), and the claim follows.

From the above two lemmas and ( 22), we have the asymptotic

for any b 1 and ζ ∈ Ω, where w 0 = w 0 (b, ζ ) is the quantity in Lemma 6. Now we can control sum (18). As before, we assume that z = xi y, where y = κ log + x for some C κ C. From ( 18), one has

where Q t,n is the quantity

We first consider the estimation of Q n in the main case when n is not too huge, in the sense that n x exp 100

In this case, if we apply Lemma 6(ii) with ζ = 9 + y + i x and b = πn 2 , we have that the quantity w 0 = w 0,t,n arising in that lemma obeys the asymptotics

which when combined with (23) gives

In particular, the factor 1 √ be 4w 0

and thus by (31),

where we have again used (23). From (34) (and using t = O(1) and y = O C (log + x) to bound some small error terms), we can calculate the quantity Re

and thus on cancelling and gathering terms, we obtain

where J t = J t (x) and K t,n = K t,n (x) are the positive quantities

and

A similar computation gives

In particular, we have the upper bound

for 1 n x exp(100 x 1/2 |t| ), and for n = 1, we have the refinement

x 4π 9+y 4

Using the crude bound

Also, from (35), we have

Thus, to finish the proof of (8) (or ( 16)), one just needs to show that the tail

. Suppose now that n > x exp(100 x 1/2 |t| ). If we now apply Lemma 6(iii) with ζ = 9 + y + i x and b = πn 2 , and write w 0 = α + iβ with 0 < β < π/8, we have that α is negative with -α 1 8 log n, while from (31) and ( 23) (and Lemma 6(i)), we have

and similarly for I t (πn 2 , 5+ y +i x). Since log n 100

which already gives (7). Sending t to 0, taking absolute values, and then dividing using (36) and (35), we obtain after cancelling all the t-independent terms that

Since the ratio H t H 0 is holomorphic in the region of interest, we can thus find a holomorphic branch of log

for all T 0, where we use Ψ : R + → R to denote the function

(It is traditional to also insert the lower order term -7 8 here, but this term will not be of use in our analysis and will therefore be discarded. The factors of 4π are not of particular significance and may be ignored by the reader on a first read.) For future reference, we record the derivative of Ψ as

in particular, Ψ is increasing for T > 4π. Applying (37) with T replaced by T + α and subtracting, we conclude from the mean value theorem that

for all T 0 and 0 α C for any fixed C, where the implied constants in the asymptotic notation are allowed to depend on C. Because we are assuming the Riemann hypothesis (and hence the Lindelöf hypothesis), one can improve this latter bound to

n s that is valid in this region. Define the classical location ξ j of the jth zero for j 1 to be the unique quantity in (1, +∞) solving the equation

and extend this to negative j by setting ξ -j := -ξ j . (As with the quantity w 0 introduced in Lemma 6, one could express ξ j explicitly in terms of the Lambert W function if desired as ξ j = 4π e exp(W ( j/e)), but we will not use this relation in this paper.) Clearly, ξ j are increasing in j. For future reference, we record the following bounds on ξ j .

LEMMA 8 (Spacing of the classical locations).

(i) For any j 1, one has

In particular, ξ j j log + j and log + ξ j log + j. (ii) For any j, k ∈ Z * , one has

(iii) If 1 j k, then one has the more precise approximation

Of course, the implied constant in the error term in (45) can depend on the implied constants in the hypothesis j k.

Proof. If j 1, then from (38), one has

which implies that j 1/2 ξ j j (say), which implies that log + ξ j log + j; substituting this back into (46) yields

.

This in turn implies that log + ξ j = (1 + o j→∞ (1)) log + j, and using (46) one last time gives (42). Now we obtain (ii). If j, k have opposing sign, then (44) follows from (43), so by symmetry we may assume that j, k are both positive. If j is much larger than k or vice versa, then bound (44) follows from (43) and the triangle inequality, so we may now restrict attention to the case 1 j k. Estimates (44) and (45) are trivial for j = O(1), so we may assume j to be large.

From (42), we have

and hence by the mean value theorem and (39), we have

for some T between ξ k and ξ j . From (43), we see that T ξ j , and so (44) follows. Furthermore, we can conclude that

Applying (37) to T = x j (0) for some j 1, we conclude in particular that

). From (39) and the mean value theorem, we conclude that

for all j 1, and hence for all j ∈ Z * by symmetry. (One may wish to treat the bounded case j = O(1) separately to avoid the minor issue that Ψ (T ) becomes decreasing for T < 1.) In particular, from (43) and the fact that x 1 (0) > 0, we conclude that x j (0) j log + ξ j j log + j for all j 1.

In a similar vein, if 1 j < k j + log + j, then from applying (41) with T = x j (0) and α equal to (or slightly less than) x k (0) -x j (0), we have (1).

Informally, this asserts that the zeros x j (0) behave like an arithmetic progression of spacing 4π log + ξ j at spatial scales between o(1) and 1. (In fact, when combined with (47) and (45), we see that this behaviour persists for all scales between o(1) and o(ξ j ).)

In this section, we use the asymptotics on H t obtained in the previous section to establish analogous, but weaker, bounds for the zeros x j (t) of the functions H t , in which we lose an additional logarithm factor in the error estimates. THEOREM 9 (Riemann-von Mangoldt type formulae). Let Λ < t 0, T > 0, and let 0 α C for some C > 0. Then one has

and

The decay rate in the o T →∞ () error term is permitted to depend on C but is otherwise uniform in α.

Repeating the previous analysis, we conclude the following.

COROLLARY 10 (Macroscopic structure of zeros). Let Λ < t 0. Then one has

for all j ∈ Z * ; in particular,

for all j 1. We also have

Informally, this corollary asserts that the zeros x j (t) behave like an arithmetic progression of spacing 4π log + ξ j at spatial scales between o(log + ξ j ) and o(ξ j ). This level of spatial resolution is worse by a factor of log + ξ j than what one can achieve for x j (0), but will still (barely) be enough for our applications. We remark that a significantly sharper estimate (with an error term of just O(1) in the analogue of (48)) is available for any fixed t > 0; see [13,Theorem 1.4]. (In press: even sharper estimates have recently been obtained in [21,Theorem 1.5].)

We now turn to the proof of the two bounds in Theorem 9.

Proof of (48). We make use of the argument principle in exactly the same manner as in the classical proof of the Riemann-von Mangoldt formula. By perturbing T slightly if necessary, we may assume that T is not a zero of H t .

Let κ > 0 be a sufficiently large absolute constant. Then the argument principle yields

where Γ is the counterclockwise contour carved out by a straight line from iκ log + 0 = iκ log 2 to -iκ log + 0 = -iκ log 2, then along the curve Γ I parameterized by xiκ log + x for x ∈ [0, T ], then along the line Γ I I from Tiκ log + T to T , then along the vertical line conjugate to Γ I I and the curve conjugate to Γ I , leading back to i log 2. As the integrand is odd, the integral along the line from iκ log + 0 to -iκ log + 0 vanishes. Using the symmetry H t (z) = H t (z), we thus have

From ( 9) and (39), one sees that 1 π

for z = x-iκ log + x on Γ I (extending Ψ to the right half-plane using the standard branch of the logarithm), and hence by the fundamental theorem of calculus

). On the other hand, if we let θ be a phase so that e iθ H t (Tiκ log + T ) is real and positive, then

where m is the number of zeros of Re e iθ H t (z) along the contour Γ I I , since the left-hand side is the change in arg e iθ H t (z) as z varies over this contour, and for each increment of π in the value of arg e iθ H t (z), we must have that Re e iθ H t (z) is zero for some z. Note that the number of zeros of ReH t (z) along this contour is the same as the number of zeros of g(s) := 1 2 (e -iθ H t (is + T ) + e iθ H t (-is + T )) as s ranges along the line from 0 to κ log + T . Hence m is no more than the number of zeros m of g(s) in the disc of radius κ log + T centred at κ log + T . We can estimate the count m with the Jensen formula as follows. Let M be the maximum of g(s) in a disc centred at κ log + T of radius 2κ log + T . Using (7) and the conjugate symmetry of H t (z), we have

T +O(log 2 + T ) .

Since from (8) we have g(κ log

T +O(log 2

+ T ) , it therefore follows from the Jensen formula (see for example, [16, Lemma 6.1]) that m log 2 + T. This induces a corresponding bound on the integral of Proof of (49). We will use a 'limiting profile argument' (also known as a 'compactness argument' or 'normal families argument'), in which one extracts and then studies a limit of suitably rescaled versions of a family of analytic functions to conclude asymptotic information about these functions. We remark that this sort of argument can also be used in a similar fashion to deduce the Lindelöf hypothesis from the Riemann hypothesis: see Theorem 1 of terrytao.wordpress.com/2015/03/01.

Suppose for contradiction that this claim failed; then there exists a sequence T n → ∞, and bounded sequences Λ < t n 0 and 0 α n C, as well as an ε > 0, such that

for all n. By perturbing T n slightly, we may assume that H t n does not vanish at T n or T n + α n . Let κ > 0 be a sufficiently large absolute constant. By the hypothesis Λ < t n , the function H t n has no zeros in the lower half-plane. Thus we can define holomorphic functions F n on the lower half-plane by the formula

with the branch of the logarithm chosen so that F n (-iκ) = 0. From (7), we see that F n are uniformly bounded on any compact subset of the lower half-plane. Thus, by Montel's theorem (see [24, Section 3.2]), we may pass to a subsequence and assume that F n converge locally uniformly to a holomorphic function F on the lower half-plane; since F n all vanish on -iκ, F does also. Then by the Cauchy integral formula, the derivatives

converge locally uniformly to F . Comparing this with (9), we conclude that

whenever the imaginary part of z is sufficiently large and negative. By unique continuation, we thus have F (z) = 1 4 for all z in the lower half-plane; as F vanishes on -iκ, we thus have

on the lower half-plane. Since F n converges locally uniformly to F, we conclude that

54) uniformly for z in a compact subset of the lower half-plane. Similarly, since F n converges locally to F, we have

uniformly for z in a compact subset of the lower half-plane. Let δ > 0 be a small constant. As in the proof of (48), we can use the argument principle (and a rescaling) to write

where Γ I,n , Γ I I,n , Γ I I I,n trace the line segments from 0 to -iδ, from -iδ to α n -iδ and from α n -iδ to α n , respectively. By (55), the contribution of the Γ I I,n integral is α+o n→∞ (1)+O(δ) 4π log 2 + T n (we allow the decay rate in the o n→∞ (1) errors to depend on δ). Applying the Jensen formula argument used to prove (48), we see that the contribution of the Γ I,n integral is bounded in magnitude by

where

and the phase θ n is chosen so that e iθ n H t n (T n -iδ log + T n ) is real and positive. Applying (54) (and the functional equation

say), and (7) (and the functional equation) otherwise, we conclude that the Γ I,n integral is equal to

We have this similarly for the Γ I I I,n integral. Taking δ to be sufficiently small and n sufficiently large, we contradict (53).

Dynamics of zeros

As remarked in Section 1, the functions H t solve a backward heat equation. As worked out in [11], this induces a corresponding dynamics on the zeros x j of H t . THEOREM 11 (Dynamics of zeros). For Λ < t 0, the zeros x j (t) depend in a continuously differentiable fashion on t for each j, with the equations of motion

for k ∈ Z * and Λ < t 0, where recall that the tick denotes principal value summation over j ∈ Z * (which will converge thanks to (50) and ( 43)).

Proof. This follows from [11,Lemma 2.4] (the continuity of the derivative following for instance from [11, Lemma 2.1]).

Informally, ODE (56) indicates that the zeros x k (t) will repel each other as one goes forward in time. On the other hand, if x k (t) are arranged (locally, at least) in an arithmetic progression, then ODE (56) suggests that the zeros will be in equilibrium. If x k are not arranged in an arithmetic progression, and instead have some fluctuation in the spacing between zeros, then heuristically ODE (56) suggests that the zeros would move away from the more densely spaced regions and towards more sparsely spaced regions, thus converging towards the equilibrium of an arithmetic progression. This is the intuition behind the convergence to local equilibrium mentioned in Section 1.

One can estimate the speed of this local convergence to equilibrium by the following heuristic calculation. Consider the zeros in a region [T, T +α] of space, where T > 0 is large and α is reasonably small (for example, α = O(log + T )). From Theorem 9 (or (43) and ( 50)), we see that we expect about α 4π log T zeros in this interval, with an average spacing of 4π log + T . Suppose for the sake of informal discussion that there is some moderate fluctuation in this spacing, for instance, suppose that the left half of the interval contains about 1.5 α 8π log T zeros and the right half contains only about 0.5 α 8π log + T zeros. Then a back of the envelope calculation suggests that for x k (t) near the middle of this interval, the right-hand side of (56) would be positive and have magnitude

Since the length of the interval is α, one may then predict that the time needed to relax to equilibrium is about α/ log + T . Since we can evolve the flow for time |Λ| 1, one would expect to attain equilibrium at the final time t = 0 if the initial length scale α of the fluctuation obeys the bound α = o T →∞ (log + T ). Happily, this upper bound is precisely what asymptotic (52) gives, so we heuristically expect to (barely) be able to establish local equilibrium at time t = 0.

Of course, one has to make this intuition more precise. Our strategy for doing so involves exploiting the formal gradient flow structure of ODE (56). (This strategy was loosely inspired by the work of Erdős, Schlein and Yau [12] exploiting the Hamiltonian structure of Dyson Brownian motion to obtain local convergence to equilibrium since the equations for Dyson Brownian motion resemble that in (56) but with an additional Brownian motion term. Indeed, Dyson Brownian motion is the diffusion related to the Gibbs measure 1 Z e -βH for the Hamiltonian studied here.) Indeed, one may formally write (56) as the gradient flow

where H is the formal 'Hamiltonian' H((x j ) j∈Z * ) := j,k∈Z * : j =k log 1

, where we ignore for this nonrigorous discussion the fact that the series defining H is not absolutely convergent. The Hamiltonian is convex, so one expects the quantity H(t) := H((x j (t)) j∈Z * ) = j,k∈Z * : j =k H jk (t)

to be decreasing and convex in time, and for the state (x j (t)) j∈Z * to converge to a critical point of the Hamiltonian, where

denotes the Hamiltonian interaction between x j (t) and x k (t). Indeed, a formal calculation using (56) yields the identity

where E is the 'energy'

denotes the 'interaction energy' between x k (t) and x k (t), and we once again ignore the issue that the series is not absolutely convergent. A further formal calculation using (56) again eventually yields

suggesting that H(t) and E(t) are decreasing and that H(t) is convex, as claimed.

In order to deal with the divergence of the infinite series appearing above, we will need to truncate the Hamiltonian and energy before differentiating them. The following lemma records some of the identities that arise when doing such truncations.

LEMMA 12 (Identities). For brevity, we suppress explicit dependence on the time parameter t ∈ (Λ, 0]. Let K ⊂ Z * be a finite set of some cardinality |K |. All summation indices such as i, j, k are assumed to lie in Z * .

(i) (Dynamics of a gap, cf. [11, Lemma 2.4]) If j, k ∈ Z * are distinct, then

.

(ii) (Cross-energy inequality, cf. [11, Lemma 2.5]) One has

in the weak sense that k∈K ; j ∈K E jk (t 2 ) -E jk (t 1 ) -

(iv) (Virial identity) One has

(The terminology here is in analogy with the virial identity in N -body classical gravitational physics; see for example, [27, Exercise 1.48].)

A key point in identities (iii), (iv) and (v) is that if one ignores the 'cross terms' involving interactions between indices in K (representing some 'local subsystem' of particles) and indices outside of K (representing the 'environment' that subsystem interacts with), the right-hand side has a definite sign (negative in the case of (iii) and (v) and positive in the case of (iv)). This gives a number of useful 'monotonicity formulae' as long as cross terms are under control. As discussed above, many of these various monotonicity formulae reflect the formal convexity properties of the Hamiltonian H. With more effort, one can obtain a precise formula for the defect in the inequality in (ii); see [11,Lemma 2.5].

Proof. From (56), one has

which gives (i). Note that the series is now absolutely convergent thanks to (43) and ( 50). Now we prove (ii). By monotone convergence, it suffices to show that

for all Λ < t 1 t 2 0 and all sufficiently large R. By the fundamental theorem of calculus, it suffices to show that

We can expand the left-hand side as

and so it will suffice to show that

If R is large enough that [-R, R] Z * contains k, we can split this sum into three parts, depending on whether

The contribution of the case i ∈ K can be rewritten as

after symmetrizing in k and k , which is clearly non-negative. Similarly the contribution of the case i ∈

which is also clearly non-negative. Finally, for i ∈ [-R, R] Z * , all summands are already non-negative. This gives (ii).

For (iii), we can similarly expand the left-hand side as

which by (i) becomes

To prove (iii), it thus suffices to establish the identity

The left-hand side expands as

.

The final sum can be rewritten as

The denominator is a Vandermonde determinant and is totally antisymmetric in k, k , k , k . All the monomials appearing in the numerator disappear upon antisymmetrization, so the final sum vanishes. To conclude the proof of (iii), it suffices to show that

The difference between the left-hand and right-hand sides can be written as

The denominator is totally symmetric in k, k , k , while the numerator symmetrizes to zero, giving the claim. Now we prove (iv). The left-hand side expands as

which by (i) becomes

.

It will thus suffice to show that

But the left-hand side can be written as

The denominator is totally antisymmetric in k, k , k . The numerator antisymmetrizes to -(

Finally we prove (v). The left-hand side expands as

It thus suffices to show that the expression

vanishes. But the summand antisymmetrizes to zero, giving the claim.

A weak bound on gaps

In order to analyse (truncated versions of) the Hamiltonian H(t) = j =k H jk (t), we will need some upper bounds on the individual terms H jk (t). It was shown in [11, Corollary 1] that these quantities are finite (that is, the zeros are simple) when Λ < t 0. It turns out that by refining the analysis in [11] (and by narrowing the range of times t to the region Λ/2 t 0), one can establish a more quantitative lower bound.

PROPOSITION 13 (Lower bound on gaps). For any j ∈ Z * and any Λ/2 t 0, one has max

The bound in (59) is probably not optimal, but for our application any bound that grows more slowly than (say) | j| 0.1 as j → ∞ would suffice.

To prove this proposition, we first need the following variant of a result in [11].

LEMMA 14. Let K be a finite subset of Z * of cardinality |K | 2, and let Λ/2 t 0.

.

Informally, this lemma asserts that the gaps within K cannot be too small, unless there is also a small gap between an element of K and an element outside of K . The strategy will be to iterate this observation to show that a very small gap will therefore propagate until it contradicts (52).

Proof. Let A = A(t) and B = B(t) denote the functions

The function A(t) is continuously differentiable. The corresponding claim for B(t) is not obvious; however, the sum defining B(t) is uniformly convergent (thanks to (51)) and hence B(t) is at least continuous. From Lemma 12(ii), we have the lower bound

(cf. [11,Lemma 2.5]) in the weak sense for Λ < t 0. In particular, if there exists a time Λ < t -< t such that sup

which rearranges as

.

By continuity, we conclude that B(t ) cannot attain or exceed the value 2B(t) anywhere in the interval (-Λ, t] ∩ (t -1 8B(t) , t), that is to say that

By hypothesis, this is a range of size at least

.

On the other hand, for t in the above range, we see from Lemma 12(iv) that

and hence by Gronwall's inequality, one has

giving the claim. Now we fix a time Λ/2 t 0 and drop the dependence on t. For any finite set

and so from the above lemma, we have

In particular, if δ(K ) c|K | -5/2 for a sufficiently small absolute constant c > 0, then we have

and hence by the pigeonhole principle, there exists k ∈ K such that

From ( 52) and (58), we have We conclude that if δ(K ) c|K | -3 for a sufficiently small c > 0, then there exists k ∈ K such that

or equivalently min

and thus (assuming that δ(K ) c|K | -3 ) we have

In either case, we call K an enlargement of K . Now we can prove Proposition 13. By symmetry, we may assume j is positive. We can also assume j is large, as the claim follows from compactness for bounded j. As before, we suppress the dependence on t. It thus suffices to show that log 1

(log 2 j) log log j for large positive j. By iterating (60) at most log j times starting from the interval K 1 := [ j, j + 1] Z * , we can find a sequence

+ ξ j with the following properties:

Since |K i | r + 1 log 2 + ξ j log 2 j and k +,i j + r j, we have from property (i) that δ(K i+1 ) j log 2 jδ(K i )

for all 1 i < r , and hence δ(K r ) exp(O(log 2 j log log j))δ(K 1 ).

On the other hand, from property (ii), using the bound |K r | r + 1 log 2 ξ j in the first case and (52) and the pigeonhole principle in the second case, we have δ(K r ) log -6 ξ j log -6 j.

Combining the two estimates, we obtain the claim.

A weak bound on integrated energy

In addition to truncations of the Hamiltonian, we will also need to control truncations of the energy j =k E jk (t). Although Proposition 13 provides some control on the summands here, it is too weak for our purposes (being of worse than polynomial growth in j, k), and we will need the following integrated bound that, while still weak, is at least of polynomial growth.

PROPOSITION 15 (Weak bound on integrated energy). Let J > 0. Then

We will use this bound to justify an interchange of a derivative and an infinite series summation in the next section.

Proof. We may take J to be large, as the claim is trivial for J in the compact region J = O(1). For any discrete interval I , let Q I denote the quantity

From (52), we have a crude lower bound

while from Proposition 13 we have an extremely crude upper bound

The ratio between Q [0.5J,3J ] Z * and Q [J,2J ] Z * is thus less than (1+ J -0.1 ) 0.5J/J 0.1 . By the pigeonhole principle, we can then therefore find an interval

where K := [J --J 0.1 , J + + J 0.1 ] Z * is a slight enlargement of K . Next, we apply Lemma 12(v) and use the fundamental theorem of calculus to obtain the identity

dt.

From Proposition 13, the left-hand side is O(J 2 log O(1) J ); thus

dt.

Using ab a 2 + b 2 , we thus have

Using (52), the contribution to the integral of those j outside of K may be crudely bounded by O(J 2 log O(1) J ) (in fact, one can improve this bound to O(J log O(1) J ) if desired, although this will not help us significantly here). The contribution of those j inside K may be bounded by

thanks to (61). We conclude that

, and the claim follows.

Strong control on integrated energy

As discussed previously, the strategy to establish convergence to local equilibrium is to study (a suitable variant of) the formal Hamiltonian

terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fmp.2020.6 Downloaded from https://www.cambridge.org/core. IP address: 107.190.7.212, on 25 Aug 2020 at 14:47:54, subject to the Cambridge Core and its derivatives, with the intention of controlling (suitable variants of) integrated energies such as 0 Λ/4 j,k∈Z * : j =k E jk (t) dt.

Unfortunately, even with the bound just obtained in Proposition 13, the above expression is far from being absolutely convergent. To address this issue, we need to mollify and renormalize the Hamiltonian and the energy in a number of ways. We renormalize the inverse square function x → 1 |x| 2 for x = 0 that appears in the definition of the energy interactions E jk (t) by introducing the modified potential

x 2 is convex, V is non-negative, and one can verify the asymptotics

(62) For any distinct j, k and any Λ/2 t 0, we define the renormalization

of the interaction energy E jk (t); we observe that

For any discrete interval I ⊂ Z * , we define the renormalized energy

this is clearly a non-negative quantity that is nondecreasing in I . It can also be simplified up to negligible error as follows.

LEMMA 16. If I = [I -, I + ] Z * is a discrete interval and Λ/2 t 0, then

Proof. By symmetry and the triangle inequality, we may assume without loss of generality that 0 I -I + ; we may then assume that I + is large, as the claim is trivial for I + in the compact region I + = O(1). By (63), it suffices to show that j,k∈I : j =k

We may desymmetrize the left-hand side as

By (50), it thus suffices to show that

Consider the inner sum k∈I : k = j 1 (ξ k -ξ j ) 3 . From (44), we see that the contribution to this inner sum of those k with |k -j|

. For the remaining range |k -j| <foot_1 2 j, we can use (45) to estimate

and so on summing we obtain

As k → 1 k-j is odd around j, the sum on the right-hand side can be estimated as O( THEOREM 17. For any T > 0, one has

The remainder of this section is devoted to a proof of Theorem 17. The claim is trivial for T in any compact region T = O(1), so we may assume without loss of generality that T is large. Recall the notation X Y or X = Õ(Y ) for X Y log O (1) T introduced in the notation section of the paper; this will be convenient to use in the argument that follows. (Typically, when we use this notation, we will also have some sort of power gain T -c that will safely absorb all the log O(1) T factors.) Let ψ T : Z * → R + be the weight function

This is a smooth positive weight that is mostly localized to the region j = O(T log T ) and fairly rapidly decaying away from this region. We introduce the smoothly truncated renormalized energy

for Λ/2 t 0. This is clearly non-negative, and from Proposition 15, (66), (62) and Fubini's theorem, we see that ẼT is absolutely integrable in time (in particular, it is finite for almost every Λ/2 t 0). Since E jk are non-negative, we see from (66) that to prove (65) it will suffice to show that 0

We have an analogue of Lemma 16.

LEMMA 18. For almost every Λ/2 t 0, one has

Proof. For almost every t, one sees from Proposition 15, (66) and Fubini's theorem (and (51)) that the series

is absolutely convergent. Thus, by Fubini's theorem and (63), it will suffice to show that j,k∈Z * :

We may desymmetrize the left-hand side (again using Fubini's theorem) as

and so it will suffice to establish the bound

As in the proof of Lemma 16, we see from (44) that the contribution of those k with |k -j| 1 2 j is acceptable. For the remaining range |k -j| < 1 2 j, we again use (45) to estimate

and similarly

and the claim follows by direct computation using the fact that k → 1 k-j is odd around j.

Recall that two indices j, k ∈ Z * are said to be nearby, and we write j

This is clearly a symmetric relation.

Next, for Λ/2 t 0, we define the smoothly truncated renormalized Hamiltonian

From (66), Proposition 13 and (45), we see that the sum here is absolutely convergent for every Λ/2 t 0. We can also express it in terms of nonnegative quantities plus a small error, in a manner similar to Lemma 18, as follows. We first introduce the renormalization

of the logarithm function x → log 1 |x| ; this is a convex non-negative function on R\{0} that vanishes precisely when |x| = 1 and obeys the asymptotics

(70) For any Λ/2 t 0 and distinct j, k ∈ Z * , we define the normalization

of the Hamiltonian interaction H jk (t); this is symmetric in j, k and non-negative, vanishing precisely when x k (t) -x j (t) = ξ k -ξ j .

LEMMA 19. For every Λ/2 t 0, one has

Proof. From (71), one has

so by (69) it suffices to show that j,k∈Z * : j∼ T k

Note from (43), (45), ( 51) and (66) that the sum here is absolutely convergent. Desymmetrizing, it suffices to show that The inner sum can be crudely bounded by Õ(1) for all j thanks to (66) and (45). By (43), ( 50) and (66), it thus suffices to show that

whenever T 0.5 | j| T 1.5 (say). For j ∼ T k, one has ψ T (k) = ψ T ( j) + Õ(T -0.8 ), and the contribution of the error term is acceptable by (45), so it suffices to show that

whenever | j| T 0.5 . But from (45), we have

As k → log ξ j 4π 1 j-k is odd around j and the set {k : j ∼ T k} is very nearly symmetric around j, it is then easy to establish (73) as required.

In contrast to the non-normalized interaction H jk (t), the quantity Hjk (t) is well controlled when k and j are far apart.

LEMMA 20 (Long-range decay of Hjk ). Let j, k be distinct elements of Z * , and let t be in the range Λ/2 t 0. There exists a quantity ε( j) that goes to zero as | j| → ∞, such that if |k -j| ε( j) -1 log 2 + ξ j , then

, and if ε( j) log 2 + ξ j |k -j| ε( j) -1 log 2 + ξ j , one has the refinement

Finally, in the remaining region |k -j| < ε( j) log

By symmetry, we can take j positive; we may also assume j to be large, as the bounded case j = O(1) may be treated by compactness. From (50), one then has

and from (44), one has

and hence

The claim then follows from (70). Next, suppose that ε( j) log 2 + ξ j |k -j| ε( j) -1 log 2 + ξ j . In this case, from (52) (iterated O(ε( j) -1 ) times) and (45), we have

while from (44) we continue to have (75), and hence

with the decay rate in the o j→∞ notation independent of the choice of function ε(). For ε( j) going to zero sufficiently slowly, the claim once again follows from (70). Finally, for the remaining case |k -j| < ε( j) log 2 + ξ j (which implies x k (t)x j (t) log 2 + j thanks to (52)), the claim follows from Proposition 13 and (70).

is moderately sized.

(ii) The quantity

is moderately sized.

(iii) For any absolute constants C, c > 0, the expression

is negligible.

(iv) For any absolute constants C, c > 0, the expression

We have this similarly if x i (t) are replaced by ξ i throughout.

Proof. For brevity, we omit the explicit dependence on the time t. Also, all summation indices i, j, k are understood to range in Z * . From (44), we see that

for all j ∈ Z * , and hence

From this and Lemma 16, we conclude (i). Using

we then obtain (ii). If instead we use

we obtain (iii) and (iv). We have this similarly if x i are replaced by ξ i throughout.

We now have the following crucial derivative computation.

PROPOSITION 22. In the range Λ/2 t 0, the function H T is absolutely continuous, and the derivative ∂ t HT (t) is equal to -4 ẼT (t) plus negligible terms for almost all t. In other words, one has

for almost every t.

REMARK 23. This may be compared with Lemma 12(v) or indeed the formal identity (57). That the right-hand side is approximated in terms the renormalized energy, rather than just the energy, may be thought of heuristically as being a result of ∂ t H vanishing when the zeros x j settle on an equilibrium, being spaced like the points ξ j .

Proof. As before, we omit the explicit dependence on t, and all summation indices are understood to lie in Z * . By (56), we have

If we formally insert this into (69) and desymmetrize in j and k, we would obtain the identity

However, we need to justify the interchange of the derivative and the infinite summation. First, we use the fundamental theorem of calculus to rewrite (77) in integral form as

for any Λ/2 t 0 0. Multiplying by ψ T ( j)ψ T (k), we conclude that

the dominated convergence theorem, we can interchange the outer sum and the integral as soon as we can show that the expression

dt is finite. By symmetry in j and k, it suffices to show that

is finite. But using (50) and (52), we can crudely bound

t)| (using the convention x 0 (t) = 0), so expression (79) may in turn be crudely bounded by and this will be finite thanks to Proposition 15 and (66). We conclude (after desymmetrizing in j and k) that

The above analysis also shows that the integrand is absolutely integrable in time.

From the Lebesgue differentiation theorem, we conclude that HT is absolutely continuous and that (78) holds at almost every time t.

To conclude the proof of the proposition, it will thus suffice to show that

is equal to ẼT plus negligible terms. We can split this expression as X 1 + X 2 + X 3 + X 4 , where

We first claim that X 4 is negligible. From (50), we have

From (44), we may crudely bound this sum by Õ(1). By Lemma 21(iii), this shows that the contribution to X 4 of those k for which |k| T 0.9 or |k| T 1.1 (say) is negligible, so we may assume T 0.9 |k| T 1.1 . Let A 2 be a large constant. Using (44), we may write

For the first sum on the right-hand side, we use (45) (as in the proof of (74)) as well as (43) to conclude that

where the subscript in the O A notation means that the implied constant can depend on A.

Meanwhile, combining the i and -i terms and using ( 43) and (44), we have

Sending A slowly to infinity, we conclude that

and the negligibility of X 4 then follows from Lemma 21(ii). Now we claim that X 2 is negligible. Thanks to the restrictions on i, j, k, we see that

and hence

symmetrizes to zero, and hence

and performing the i or j summation respectively, we conclude that

and so X 2 is negligible thanks to Lemma 21(i).

We have shown that expression (80) is equal to X 1 + X 3 plus negligible terms. A similar argument (replacing x i with ξ i throughout) shows that the expression

is equal to X 1 + X 3 plus negligible terms, where

From Lemma 18, we see that ẼT is equal to

up to negligible terms. From ( 43) and (44), we have

when j = k and j ∼ T k, so the final term in (82) is negligible. Thus, to complete the proof of the proposition, it will suffice to show that expression (81) and the difference X 3 -X 3 are both negligible. Expression (81) may be rearranged as

By (44), both inner sums are Õ(1), so the contribution of those |k| T 0.5 or |k| T 1.5 (say) is negligible. For T 0.5 < |k| T 1.5 , we see from (72) that the factor j: j∼ T k ψ T ( j) ξ k -ξ j is o T →∞ (log T ), and from (44) and (73) and the triangle inequality, we also see that i: i =k 1 ξ k -ξ i = O(log T ). Thus (81) is negligible as required.

Finally, we show that X 3 -X 3 is negligible. This quantity may be written as

Observe that if |k -j| and |k -i| are both larger than or equal to T 0.1 , then from (44) and ( 50), one has

and so the contribution of this case is negligible. From the triangle inequality, we see that it is not possible for |k -j| and |k -i| to both be less than T 0.1 , so it remains to treat the components

Consider first (83). From the triangle inequality, we have | j -k| T 0.2 , and hence by (50),

By Lemma 21(ii) and (44), we may thus replace 1 x k -x j by 1 ξ k -ξ j at negligible cost in (83), leaving us with i, j,k: 0<| j-k|<(T 2 +| j|+|k|) 0.1 0<|i-k|<T 0.1

; |i-j|>(T 2 +|i|+| j|) 0.1

up to negligible errors. But by (45) and the hypothesis |i -k| T 0.1 , one may bound

when T 0.9 |k| T 1.1 , and use the weaker bound

for all other k, so this expression is also negligible by Lemma 21(ii), (iii), (iv) (noting that ψ T (k) and ψ T (i) are comparable). A similar argument also handles (84).

To use Proposition 22, we need estimates that ensure ẼT is large when HT is large. To this end, we have the following lemma.

LEMMA 24. Let m be a natural number, and let Λ/2 t 0. Let T > 0, and let δ = δ(T ) go to zero as T → ∞ sufficiently slowly. If HT (t) δmT log 3 + T , then ẼT (t) δ2 2m T log 3 + T , where the implied constant is absolute.

Proof. As before, we suppress explicit dependence on t, and we may assume T to be large as the claim is trivial from compactness for T = O(1). From Lemma 19, we have (for δ decaying sufficiently slowly) that ψ T ( j)ψ T (k) Hjk (t) δmT log 3 T.

But for j, k in this sum, we see from ( 62) and (70) that

and the claim follows.

We can now shrink HT down to a reasonable size in finite time. Proof. We may take T to be large. From Proposition 22 and Lemma 24, we see that for any natural number m, and for almost every time t for which one has HT (t) δmT log 3 T, one has ∂ t HT (t) -cδ2 2m T log 3 T for some absolute constant c > 0. In particular, if m is larger than some large absolute constant m 0 , and Λ/2 t Λ/4 is such that δmT log 3 T HT (t) δ(m + 1)T log 3 T,

then it is not possible (for m 0 large enough) to have HT (t ) δmT log 3 T for all t t t + c -1 2 -2m , as this would violate the fundamental theorem of calculus for absolutely continuous functions. Thus, by the intermediate value theorem, there exists t t t + c -1 2 -2m such that δ(m -1)T log 3 T HT (t ) δmT log 3 T, and on iterating this we conclude (for m 0 large enough) that there exists t t t + 2c -1 2 -2m 0 such that HT (t ) δm 0 T log 3 T.

We run this argument with t set equal to Λ/2, and m the unique integer obeying (88), to conclude (for m 0 large enough) that there exists Λ/2 t Λ/4 obeying (89). (Note that this conclusion is immediate if the initial value of m was already less than m 0 .) On the other hand, from Proposition 22, we have ∂ t HT (t) O(δT log 3 T ) for almost every t t 0 if δ decays sufficiently slowly. The claim now follows from the fundamental theorem of calculus (absorbing m 0 into the implied constants), recalling that HT is non-negative.

Controlling the energy at time 0

In the previous section, we controlled a time average of the energy. Now, using monotonicity properties of the energy, we can in fact control energy at time zero. It remains to establish Proposition 27. We use an argument to Bourgain [2, Section 4] that combines local conservation laws (or, in this case, local monotonicity formulae) with the pigeonhole principle.

The first step is to locate a good subset of particles indexed by an interval close to [I -, I + ] that does not gain too much energy due to interactions with its environment because of the separation between these particles and the environment. From (50) and the pigeonhole principle, one can find natural numbers I - j --1 < j -I -+ log 3 T I + -log 3 T j + < j + + 1 I + such that x j -(t 2 ) -x j --1 (t 2 ) 1 log T (90) (say) and similarly

x j + +1 (t 2 ) -x j + (t 2 ) 1 log T .

From Lemma 12(iv) applied to K = { j --1, j -}, we have

for all t 1 t t 2 . Since t 2 -t 1 1 100 log 2 T , we conclude from the fundamental theorem of calculus and (90) that x j -(t) -x j --1 (t) 1 log T (91) for all t 1 t t 2 . Similarly

x j + +1 (t) -x j + (t)

The basic point is that because the particles x j -, . . . , x j + never get too close to the remaining particles x j , j < j -and x j , j > j + in the system, the total energy of the former set of particles will remain approximately conserved over short periods of time thanks to Lemma 12. More precisely, let K denote the discrete interval K := [ j -, j + ], and define the un-normalized energy

From Lemma 12, we have

for t 1 t t 2 . But from (50), ( 91) and (92), we have

From the fundamental theorem of calculus, we conclude that E K (t 2 ) E K (t 1 ) + Õ(1), which by monotonicity of E K in K implies that E I (t 2 ) E I (t 1 ) + Õ(1).

Applying Lemma 16, we conclude that

But from (44), one has j∈I \I k∈I : j =k 1 (ξ j -ξ k ) 2 1, and Proposition 27 follows.

Contradicting pair correlation

It remains to see that Proposition 26 is in contradiction with results that are known to be the case for the points x j (0). Note in particular that

x j+1 (0) -x j (0) ξ j+1 -ξ j Ẽ[T log T,2T log T ] (0).

In this range using (43) and (45), we have ξ j+1 -ξ j ∼ 4π/ log + T , and so Proposition 26 implies that log 2 T T log T j, j+1 2T log T V

x j+1 (0) -x j (0) ξ j+1 -ξ j = o T →∞ (T log 3 T ).

By Markov's inequality (see [28,Ch. 1]), this implies that In particular, since the points x j (0) are twice the imaginary ordinates of nontrivial zeros of the Riemann zeta function, this implies that the gaps between the zeros of the zeta function are rarely much larger or smaller than the mean spacing. But this contradicts perhaps most strikingly the results of Montgomery [15], who determined on the Riemann hypothesis the pair correlation measure for the zeros, measured against a class of band-limited functions. As noted by Montgomery, his result implies that a positive proportion of zeros have a spacing between them strictly smaller than the mean spacing. The proof of this claim is not written down in [15], but Conrey et al. prove as their main result of [6] (using different ideas) that for any λ > .77, there exists a constant c(λ) > 0 such that at least a proportion c(λ) of j T log T satisfy x j+1 (0) -x j (0) λ 4π log T .

This contradicts (93) and therefore the assumption that Λ < 0.