# Effective approximation of heat flow evolution of the Riemann $$\xi$$ function, and a new upper bound for the de Bruijn–Newman constant

@article{Polymath2019EffectiveAO,
title={Effective approximation of heat flow evolution of the Riemann \$\$\xi \$\$ function, and a new upper bound for the de Bruijn–Newman constant},
author={D. H. J. Polymath},
journal={Research in the Mathematical Sciences},
year={2019}
}
• D. Polymath
• Published 29 April 2019
• Mathematics
• Research in the Mathematical Sciences
For each $t \in \mathbf{R}$, define the entire function $$H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du$$ where $\Phi$ is the super-exponentially decaying function $$\Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp(-\pi n^2 e^{4u} ).$$ This is essentially the heat flow evolution of the Riemann $\xi$ function. From the work of de Bruijn and Newman, there exists a finite constant $\Lambda$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are…
8 Citations

### THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

• Mathematics
Forum of Mathematics, Pi
• 2020
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

### Price ’ s Theorem , the Riemann Zeta Function and the Riemann Hypothesis

Using an extension of the Price’s theorem we construct a dynamical systems model whose steady-state converges to the Riemann’s Xi function over a restricted domain that includes the critical strip.

### A Dynamical Systems Framework for Generating the Riemann Zeta Function and Dirichlet L-functions

Using an extension of the Price’s theorem we show how to construct a dynamical systems model which in its steady-state serves as an analytic continuation of the completed Riemann zeta function and

### Will a physicist prove the Riemann hypothesis?

• M. Wolf
• Mathematics
Reports on progress in physics. Physical Society
• 2019
The Riemann Hypothesis is formulated and some physical problems related to this hypothesis are reviewed: the Polya--Hilbert conjecture, the links with Random Matrix Theory, relation with the Lee--Yang theorem on the zeros of the partition function and phase transitions, random walks, billiards etc.

### A pr 2 02 0 The Riemann hypothesis is true up to 3 · 10 12

• Mathematics
• 2020
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3 · 10. That is, all zeroes β + iγ of the Riemann zeta-function with 0 < γ ≤ 3 ·

### Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

• Physics
Sojourns in Probability Theory and Statistical Physics - I
• 2019
In the introduction to this volume, we discuss some of the highlights of the research career of Chuck Newman. This introduction is divided into two main sections, the first covering Chuck’s work in

### The Riemann hypothesis is true up to 3·1012

• Mathematics
Bulletin of the London Mathematical Society
• 2021
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3·1012 . That is, all zeroes β+iγ of the Riemann zeta‐function with 0

## References

SHOWING 1-10 OF 32 REFERENCES

### THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

• Mathematics
Forum of Mathematics, Pi
• 2020
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

### The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis

• Mathematics
Numerical Algorithms
• 2005
A new numerical method is investigated, base on the Laguerre inequalities, for determining lower bounds for the de Bruijn-Newman constant ∧, which is related to the Riemann Hypothesis, to obtain the new lower bound, -0.0991 < ∧ which improves all previously published lower bound for ∧.

### On the roots of the Riemann zeta-function

on the critical line a=l/s, t > 0 . These results confirm those made previously by Gram [1], Hutchinson [2], Titchmarsh [3] and Turing [4] and extend these to the first 10,000 zeros of \$ (s). All

### Fourier transforms with only real zeros

The class of even, nonnegative, finite measures p on the real line such that for any b > 0 the Fourier transform of exp(bt2) dp(t) has only real zeros is completely determined. This result is then

### Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

• Mathematics
• 1994
AbstractWe give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH0 (which is closely related to the Riemann ξ-function) to be a “Lehmer pair” of zeros ofH0.

### NOTES ON LOW DISCRIMINANTS AND THE GENERALIZED NEWMAN CONJECTURE

Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of qua- dratic Dirichlet L-functions. There is a real constant LKr (gen- eralizing the de

### On the exact location of the non-trivial zeros of Riemann's zeta function

• Mathematics
• 2013
In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where

### Multiplicative Number Theory

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The

### Pair correlation of zeros of the zeta function.

s T— »oo and U-+Q in such a way that UL = A; here L = ̂ —logT is the average 2n density of zeros up to T and A is an arbitrary positive constant. If the same number of points were distributed at