# 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} }

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

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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.

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

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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

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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…

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