arXiv 1807.112012018-07-30math.NT

与zeta函数零点有关的超越和

Transcendental sums related to the zeros of zeta functions

Sanoli Gun, M. Ram Murty, Purusottam Rath

备注：: to appear in Mathematika

摘要：

虽然黎曼zeta函数的非平凡零点的分布构成了数学中的一个中心主题，但对这些非平凡零点的代数性质一无所知。本文研究了$$ \sum_{ρ} R(ρ) x^ρ, $$型和的超越性质，其中$$ \sum_{ρ} R(ρ) x^ρ, $$是$ζ(s)$的非平凡零点$ρ$上的和，$R(x) \in \overline{\Q}(x) $是代数数上的有理函数，$x >0$是实代数数.特别地，我们证明了函数$$ f(x) = \sum_{ρ} \frac{x^ρ}ρ $$在$(1, \infty)$中有无穷多个零点，其中至多有一个是代数的。研究$x<1$范围内的$f(x)$所需的超越工具似乎与$x>1$范围内的工具不同。对于$x < 1$，我们有以下非零定理：如果对于一个整数$d \ge 1$，$f(π\sqrt{d} x)$在$(0,~1/π\sqrt{d})$中有一个有理数零，那么$$ L'(1,χ_{-d}) \neq 0, $$其中$χ_{-d}$是与虚二次域$K:= \Q(\sqrt{-d})$相关联的二次特征标.最后，我们考虑类似的问题，在塞尔伯格类的元素。我们的证明依赖于解析数论和超越数论的结果。

英文摘要：

While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form $$ \sum_{ρ} R(ρ) x^ρ, $$ where the sum is over the non-trivial zeros $ρ$ of $ζ(s)$, $R(x) \in \overline{\Q}(x) $ is a rational function over algebraic numbers and $x >0$ is a real algebraic number. In particular, we show that the function $$ f(x) = \sum_{ρ} \frac{x^ρ}ρ $$ has infinitely many zeros in $(1, \infty)$, at most one of which is algebraic. The transcendence tools required for studying $f(x)$ in the range $x<1$ seem to be different from those in the range $x>1$. For $x < 1$, we have the following non-vanishing theorem: If for an integer $d \ge 1$, $f(π\sqrt{d} x)$ has a rational zero in $(0,~1/π\sqrt{d})$, then $$ L'(1,χ_{-d}) \neq 0, $$ where $χ_{-d}$ is the quadratic character associated to the imaginary quadratic field $K:= \Q(\sqrt{-d})$. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.

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