Abstract
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$\unicode[STIX]{x1D6EC}$(thede 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$\unicode[STIX]{x1D6EC}<t\leqslant 0$, until one establishes that the zeros of$H_{0}$are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery.
Highlights
Let H0 : C → C denote the function H0(z) := 1 iz + (1)where ξ denotes the Riemann xi function ξ(s) := s(s − 1) π −s/2Γ s ζ (s) (2)and ζ is the Riemann zeta function
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’, in particular establishing a bound of the form 1 log x j+1(t) − x j (t) log2 j log log j for all large j in the range Λ/2 t 0; see Proposition 13 for a more precise statement
By Montel’s theorem, we may pass to a subsequence and assume that Fn converge locally uniformly to a holomorphic function F on the lower half-plane; since Fn all vanish on −iκ, F does
Summary
H0 is an entire even function with functional equation H0(z) = H0(z), and the Riemann hypothesis is equivalent to the assertion that all the zeros of H0 are real. De Bruijn showed that the zeros of Ht are purely real for t 1/2 Strengthening these results, Newman [17] showed that there is an absolute constant −∞ < Λ 1/2, known as the De Bruijn–Newman constant, with the property that Ht has purely real zeros if and only if t Λ. 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
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