• Corpus ID: 218581575

A New Proof of Newman's Conjecture and a Generalization

@article{Dobner2020ANP,
  title={A New Proof of Newman's Conjecture and a Generalization},
  author={Alexander Dobner},
  journal={arXiv: Number Theory},
  year={2020}
}
Newman's conjecture (proved by Rodgers and Tao in 2018) concerns a certain family of deformations $\{\xi_t(s)\}_{t \in \mathbb{R}}$ of the Riemann xi function for which there exists an associated constant $\Lambda \in \mathbb{R}$ (called the de Bruijn-Newman constant) such that all the zeros of $\xi_t$ lie on the critical line if and only if $t \geq \Lambda$. The Riemann hypothesis is equivalent to the statement that $\Lambda \leq 0$, and Newman's conjecture states that $\Lambda \geq 0$. In… 

Figures from this paper

Jensen polynomials are not a viable route to proving the Riemann Hypothesis
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THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE
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

References

SHOWING 1-10 OF 22 REFERENCES
The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis
TLDR
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 ∧.
An improved bound for the de Bruijn–Newman constant
  • A. Odlyzko
  • Mathematics, Computer Science
    Numerical Algorithms
  • 2004
TLDR
Improve previous lower bounds and prove that −2.7⋅10−9<Λ satisfies Λ≤0.7, providing yet more evidence that the Riemann hypothesis, if true, is just barely true.
Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann Hypothesis
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
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
An improved lower bound for the de Bruijn-Newman constant
TLDR
This article reports on computations that led to the discovery of a new Lehmer pair of zeros for the Riemann Ϛ function and improves the known lower bound for de Bruijn-Newman constant A.
A NEW LEHMER PAIR OF ZEROS AND A NEW LOWER BOUND FOR THE DE BRUIJN-NEWMAN CONSTANT
The de Bruijn-Newman constant has been investigated extensively because the truth of the Riemann Hypothesis is equivalent to the assertion that 0. On the other hand, C. M. Newman conjectured that 0.
On the structure of the Selberg class, I: 0≤d≤1
The Selberg class S is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global L-functions arising from number
A new lower bound for the de Bruijn-Newman constant
SummaryStrong numerical evidence is presented for a new lower bound for the so-called de Bruijn-Newman constant. This constant is related to the Riemann hypothesis. The new bound, −5, is suggested by
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