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 \(x \rightarrow \infty \).

中文翻译：

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Riemann $$ \ xi $$ξ函数的热流演化的有效近似，以及de Bruijn-Newman常数的新上限

对于每个\（t \ in \ mathbb {R} \），定义整个函数$$ \ begin {aligned} H_t（z）{：=} \，\ int _0 ^ \ infty e ^ {tu ^ 2} \ varPhi（u）\ cos（zu）\ \ mathrm {d} u，\ end {aligned} $$其中\（\ varPhi \）是超指数衰减函数$$ \ begin {aligned} \ varPhi（u） {：=} \，\ sum _ {n = 1} ^ \ infty（2 \ pi ^ 2 n ^ 4 e ^ {9u}-3 \ pi n ^ 2 e ^ {5u}）\ exp（-\ pi n ^ 2 e ^ {4u}）。\ end {aligned} $$本质上是Riemann \（\ xi \）函数的热流演化。根据de Bruijn和Newman的工作，存在一个有限常数\（\ varLambda \）（de Bruijn–Newman常数），使得\（H_t \）的零点在\（t \ ge \ varLambda \）。黎曼假设等于断言\（\ varLambda \ le 0 \） ; 最近，Rodgers和Tao建立了匹配的下界\（\ varLambda \ ge 0 \）。Ki和Kim和Lee确定了上限\（\ varLambda <\ frac {1} {2} \）。在本文中，我们针对\（t \ ge 0 \）在\（H_t（x + iy）\）上建立了一些有效的估计，其中包括一些对于x的中小值准确的估计。通过将这些估计与数值计算相结合，我们可以获得新的上限\（\ varLambda \ le 0.22 \）以及对黎曼假设进行进一步数值验证的条件。我们还获得了一些新的估计值，它们控制\（H_t（x + iy）\）的零的渐近行为为\（x \ rightarrow \ infty \）。