More Like this

1. On the number variance of zeta zeros and a conjecture of Berry

   https://doi.org/10.1112/mtk.12184  

   Lugar, Meghann Moriah; Milinovich, Micah B.; Quesada‐Herrera, Emily (January 2023, Mathematika)

   Abstract Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non‐zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non‐universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000). 

   more » « less
2. An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function

   https://doi.org/10.4064/aa230612-20-3  

   Baluyot, Siegfred_Alan C; Goldston, Daniel Alan; Suriajaya, Ade Irma; Turnage-Butterbaugh, Caroline L (April 2024, Acta Arithmetica)

   Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least 67.9% of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery’s theorem and show how to apply it to prove the following result on simple zeros: If all the zeros ρ=β+iγ of the Riemann zeta-function such that T3/8<γ≤T satisfy ∣∣β−1/2∣∣<1/(2logT), then, as T tends to infinity, at least 61.7% of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where β=1/2. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis. 

   more » « less
3. Negative discrete moments of the derivative of the Riemann zeta‐function

   https://doi.org/10.1112/blms.13092  

   Bui, Hung_M; Florea, Alexandra; Milinovich, Micah_B (May 2024, Bulletin of the London Mathematical Society)

   Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros. 

   more » « less
4. Large Deviation Estimates of Selberg’s Central Limit Theorem and Applications

   https://doi.org/10.1093/imrn/rnad176  

   Arguin, Louis-Pierre; Bailey, Emma (July 2023, International Mathematics Research Notices)

   Abstract For $$V\sim \alpha \log \log T$$ with $$0<\alpha <2$$, we prove $$\begin{align*} & \frac{1}{T}\textrm{meas}\{t\in [T,2T]: \log|\zeta(1/2+ \textrm{i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^{2}/\log\log T}. \end{align*}$$This improves prior results of Soundararajan and of Harper on the large deviations of Selberg’s Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwiłł, and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $$(\log T)^{\theta }$$, $$0<\theta <3$$, that is expected to be sharp for $$\theta> 0$$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwiłł, and one of the authors to prove fine asymptotics for the maximum on intervals of length $$1$$. 

   more » « less
5. Traces, high powers and one level density for families of curves over finite fields

   https://doi.org/10.1017/S030500411700041X  

   BUCUR, ALINA; COSTA, EDGAR; DAVID, CHANTAL; GUERREIRO, JOÃO; LOWRY–DUDA, DAVID (September 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF + 16 ] and [ Zha ]. We extend [ BDF + 16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers. 

   more » « less