素数上随机游动的一些Riemann假设
Some Riemann Hypotheses from Random Walks over Primes
摘要：
这篇文章的目的是研究各种黎曼假设如何只从素数的性质得出。为此，我们考虑两类$L$ -函数，即非主Dirichlet和那些基于尖点形式。后者最简单的例子是基于Ramanujan tau算术函数。对于这两个类，我们证明，如果一个特定的三角级数涉及的乘法字符的总和超过素数是$O(\sqrt{N})$，那么欧拉产品收敛在右半边的临界带。当这个结果与函数方程相结合时，非平凡零点被限制在临界线上。我们认为，这$\sqrt{N}$增长是一个后果的系列表现得像一个一维的随机游走。基于这些结果，我们得到了一个方程，它涉及每个个人的非平凡零的$L$ -函数的总和，包括所有的素数。最后，我们简要地提到重要的差异主要狄利克雷$L$ -函数由于极点的存在，在$s=1$，其中黎曼$ζ$ -函数是一个特殊的情况。
英文摘要：
The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is $O(\sqrt{N})$, then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this $\sqrt{N}$ growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results we obtain an equation which relates every individual non-trivial zero of the $L$-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet $L$-functions due to the existence of the pole at $s=1$, in which the Riemann $ζ$-function is a particular case.
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