AI 中文总结
本文在黎曼ζ函数所有零点为简单的假设下，给出\(\Phi(e^{-t})\)显式公式，其与经典公式有别。借此给出黎曼假设准则，还引入相关特殊整函数，表明其可表示为绝对收敛的贝塞尔函数莫比乌斯加权级数形式。
AI 中文摘要
在本文中，我们假设黎曼ζ函数的所有零点都是简单的。在此假设下，我们给出了函数\(\Phi(e^{-t})=\sum_{n=1}^{\infty}\mu(n)e^{-nt}\)的一个显式公式，它是ζ(s)和ζ′(s)在奇数整数处的值以及ζ(s)的零点的函数。该公式与经典的默滕斯函数显式公式的一个结构特征不同：\(\Gamma(s)\)的极点与ζ(s)的平凡零点碰撞，产生带有对数项的留数的双重极点。利用这个公式，我们给出了一个黎曼假设的准则：变换上的\(O(x^{-1/2})\)界无条件地蕴含黎曼假设，而相反方向需要对零点有额外假设。我们还引入了与ζ(s)相关的特殊整函数，并表明它们作为旋转自变量的贝塞尔函数的莫比乌斯加权级数允许绝对收敛的封闭形式。
英文摘要
In this paper, we assume that all the zeros of the Riemann zeta function are simple. Under this assumption we give an explicit formula for the function $Φ(e^{-t})=\sum_{n=1}^{\infty}μ(n)e^{-nt}$, as a function of the values of $ζ(s)$ and $ζ'(s)$ at the odd integers and as a function of the zeros of $ζ(s)$. A structural feature distinguishes this formula from the classical explicit formula for the Mertens function: the poles of $Γ(s)$ collide with the trivial zeros of $ζ(s)$, producing double poles whose residues contain a logarithmic term. Using this formula, we give a criterion for the Riemann hypothesis: the bound $O(x^{-1/2})$ on the transform implies the Riemann hypothesis unconditionally, while the converse direction requires additional hypotheses on the zeros. We also introduce special entire functions related to $ζ(s)$ and show that they admit absolutely convergent closed forms as Möbius-weighted series of Bessel functions of rotated argument.
Comments37 pages, includes computational appendix, 1 figure. Clearly explained non-circularity of the argument
