De Bruijn Newman Constant Research Articles

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$\unicode[STIX]{x1D6F7}$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.

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

For each $$t \in \mathbb {R}$$ , define the entire function $$\begin{aligned} H_t(z){:=}\,\int _0^\infty e^{tu^2} \varPhi (u) \cos (zu)\ \mathrm{d}u, \end{aligned}$$ where $$\varPhi $$ is the super-exponentially decaying function $$\begin{aligned} \varPhi (u){:=}\,\sum _{n=1}^\infty (2\pi ^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp (-\pi n^2 e^{4u}). \end{aligned}$$ 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 $$\varLambda $$ (the de Bruijn–Newman constant) such that the zeroes of $$H_t$$ are all real precisely when $$t \ge \varLambda $$ . The Riemann hypothesis is equivalent to the assertion $$\varLambda \le 0$$ ; recently, Rodgers and Tao established the matching lower bound $$\varLambda \ge 0$$ . Ki, and Kim and Lee established the upper bound $$\varLambda < \frac{1}{2}$$ . In this paper, we establish several effective estimates on $$H_t(x+iy)$$ for $$t \ge 0$$ , including some that are accurate for small or medium values of x. By combining these estimates with numerical computations, we are able to obtain a new upper bound $$\varLambda \le 0.22$$ unconditionally, as well as improvements conditional on further numerical verification of the Riemann hypothesis. We also obtain some new estimates controlling the asymptotic behavior of zeroes of $$H_t(x+iy)$$ as $$x \rightarrow \infty $$ .

Zen and the art of formalisation

N. G. de Bruijn, now professor emeritus of the Eindhoven University of Technology, was a pioneer in the field of interactive theorem proving. From 1967 to the end of the 1970's, his work on the Automath system introduced the architecture that is common to most of today's proof assistants, and much of the basic technology. But de Bruijn was a mathematician first and foremost, as evidenced by the many mathematical notions and results that bear his name, among them de Bruijn sequences, de Bruijn graphs, the de Bruijn–Newman constant, and the de Bruijn–Erdös theorem. The quotation above is thus interesting not because it is a reflection on his expertise in formal verification, but, rather, of his convictions as a working mathematician.

An improved lower bound for the de Bruijn-Newman constant

In this article, we report on computations that led to the discovery of a new Lehmer pair of zeros for the Riemann ζ \zeta function. Given this new close pair of zeros, we improve the known lower bound for de Bruijn-Newman constant Λ \Lambda . The Riemann hypothesis is equivalent to the assertion Λ ≤ 0 \Lambda \leq 0 . In this article, we establish that in fact we have Λ &gt; − 1.14541 × 10 − 11 \Lambda &gt; -1.14541 \times 10^{-11} . This new bound confirms the belief that if the Riemann hypothesis is true, it is barely true.

On the de Bruijn–Newman constant

If λ ( 0 ) denotes the infimum of the set of real numbers λ such that the entire function Ξ λ represented by Ξ λ ( t ) = ∫ 0 ∞ e λ 4 ( log x ) 2 + i t 2 log x ( x 5 / 4 ∑ n = 1 ∞ ( 2 n 4 π 2 x − 3 n 2 π ) e − n 2 π x ) d x x has only real zeros, then the de Bruijn–Newman constant Λ is defined as Λ = 4 λ ( 0 ) . The Riemann hypothesis is equivalent to the inequality Λ ⩽ 0 . The fact that the non-trivial zeros of the Riemann zeta-function ζ lie in the strip { s : 0 < Re s < 1 } and a theorem of de Bruijn imply that Λ ⩽ 1 / 2 . In this paper, we prove that all but a finite number of zeros of Ξ λ are real and simple for each λ > 0 , and consequently that Λ < 1 / 2 .

An improved bound for the de Bruijn–Newman constant

The Riemann hypothesis is equivalent to the conjecture that the de Bruijn–Newman constant Λ satisfies Λ≤0. However, so far all the bounds that have been proved for Λ go in the other direction, and provide support for the conjecture of Newman that Λ≥0. This paper shows how to improve previous lower bounds and prove that −2.7⋅10−9<Λ. This can be done using a pair of zeros of the Riemann zeta function near zero number 1020 that are unusually close together. The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true.

A new lower bound for the de Bruijn-Newman constant

Strong 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 high-precision floatingpoint computations, with a mantissa of 250 decimal digits, of i) the coefficients of a so-called Jensen polynomial of degree 406, ii) the so-called Sturm sequence corresponding to this polynomial which implies that it has two complex zeros, and iii) the two complex zoros of this polynomial. Aproof of the new bound could be given if one would repeat the computations i) and iii) with a floatingpoint accuracy of at least 2600 decimal digits.