More Like this

1. Strictly subgaussian probability distributions

   https://doi.org/10.1214/24-EJP1120  

   Bobkov, S G; Chistyakov, G P; Götze, F (January 2024, Electronic Journal of Probability)

   We explore probability distributions on the real line whose Laplace transform admits an upper bound of subgaussian type known as strict subgaussianity. One class in this family corresponds to entire characteristic functions having only real zeros in the complex plane. Using Hadamard’s factorization theorem, we extend this class and propose new sufficient conditions for strict subgaussianity in terms of location of zeros of the associated characteristic functions. The second part of this note deals with Laplace transforms of strictly subgaussian distributions with periodic components. This class contains interesting examples, for which the central limit theorem with respect to the Rényi entropy divergence of infinite order holds. 

   more » « less
2. Quasiconformal surgery and linear differential equations

   https://doi.org/10.1007/s11854-019-0007-9  

   Bergweiler, Walter; Eremenko, Alexandre (August 2019, Journal d'Analyse Mathématique)

   We describe a new method of constructing transcendental entire functions A such that the differential equation w″ + Aw = 0 has two linearly independent solutions with relatively few zeros. In particular, we solve a problem of Bank and Laine by showing that there exist entire functions A of any prescribed order greater than 1/2 such that the differential equation has two linearly independent solutions whose zeros have finite exponent of convergence. We show that partial results by Bank, Laine, Langley, Rossi and Shen related to this problem are in fact best possible. We also improve a result of Toda and show that the estimate obtained is best possible. Our method is based on gluing solutions of the Schwarzian differential equation S(F) = 2A for infinitely many coefficients A. 

   more » « less
3. Zero-free regions near a line

   https://doi.org/10.1007/s00209-023-03364-w  

   Bickel, Kelly; Pascoe, J E; Sargent, Meredith (November 2023, Mathematische Zeitschrift)

   We analyze metrics for how close an entire function of genus one is to having only real roots. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any poten- tial non-real zeros are forced to be farther and farther away from the critical line. 

   more » « less
4. Higher correlations and the alternative hypothesis

   https://doi.org/10.1093/qmathj/haz.043  

   Lagarias, Jeffrey C.; Rodgers, Brad (April 2020, The quarterly journal of mathematics)

   The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functionsof the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in 1/2 Z, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function. 

   more » « less
5. THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

   https://doi.org/10.1017/fmp.2020.6  

   RODGERS, BRAD; TAO, TERENCE (January 2020, Forum of Mathematics, Pi)

   For each $$t\in \mathbb{R}$$ , we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $$\unicode[STIX]{x1D6F7}$$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$ Newman showed that there exists a finite constant $$\unicode[STIX]{x1D6EC}$$ (the de Bruijn–Newman constant ) such that the zeros of $$H_{t}$$ are all real precisely when $$t\geqslant \unicode[STIX]{x1D6EC}$$ . The Riemann hypothesis is equivalent to the assertion $$\unicode[STIX]{x1D6EC}\leqslant 0$$ , and Newman conjectured the complementary bound $$\unicode[STIX]{x1D6EC}\geqslant 0$$ . In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $$\unicode[STIX]{x1D6EC}<0$$ and then analyzing the dynamics of zeros of $$H_{t}$$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $$H_{t}$$ in the range $$\unicode[STIX]{x1D6EC} 

   more » « less