Prime numbers and L-functions

Riemann Zeta函数是数学中研究最多的对象之一。1859年，德国著名数学家伯恩哈德·里曼写了他唯一一篇关于数论的文章。在这篇重要的文章中，他提出素数可以通过Riemann Zeta函数来研究。此外，他还提供了一种利用该函数研究素数的具体方法。这种方法涉及到研究被称为Riemann Zeta函数的零点的某些数字。黎曼提出了一个关于零的简单而深刻的猜想。黎曼假设是黎曼的猜想，即所有的零都位于一条直线上。这仍然是数学中最重要的悬而未决的问题之一。它激发了在不同数学领域工作的一些最优秀的研究人员的大量研究。它的解析将提供关于我们目前对素数的知识的更精细的信息。这一猜想的推广，即广义黎曼假设，也被认为是正确的。这个猜想涉及到L函数的零点的位置。L函数是Riemann Zeta函数的变种。我的研究是关于L函数及其零点的行为。这些函数还编码与素数有关的深层信息。我对素数和L函数之间的相互作用很感兴趣。此外，我的研究很大一部分是关于乘法函数的。这些函数与L函数和素数密切相关。关于L函数的零点的位置或L函数的大小的问题通常可以用乘法函数来解释。因此，我将研究某些乘法函数的更精细的行为。综上所述，一方面利用素数和乘法函数的工具研究L函数，另一方面利用L函数和乘法函数的性质来研究素数。

One of the most studied objects in mathematics is the Riemann zeta function. In 1859, the famous German mathematician Bernhard Riemann wrote his one article concerning the theory of numbers. In this important tract he proposed that prime numbers may be studied via the Riemann zeta function. In addition, he provided a specific method to study prime numbers via this function. This method involved studying certain numbers which are called the zeros of the Riemann zeta function. Riemann proposed a simple yet deep conjecture concerning the zeros. The Riemann hypothesis is Riemann's conjecture that all the zeros sit on a straight line. This remains one of the most important unsolved problems in mathematics. It has inspired a significant amount of research by some of the best researchers working in diverse fields of mathematics. Its resolution will provide much finer information concerning our current knowledge of prime numbers. A generalization of this conjecture, the Generalized Riemann Hypothesis, is also believed to be true. This conjecture concerns the location of zeros of L-functions. An L-function is a variant of the Riemann zeta function. My research concerns the behaviour of L-functions and their zeros. These functions also encode deep information concerning prime numbers. I am interested in the interplay between prime numbers and L-functions. In addition, a large part of my research concerns multiplicative functions. These functions are intimately related to both L-functions and primes. Questions concerning the location of zeros of an L-function or the size of an L-function can often be interpreted in terms of a multiplicative function. Therefore I shall investigate the finer behaviour of certain multiplicative functions. In summary, I propose to study L-functions by using tools involving prime numbers and multiplicative functions and on the other hand, I will study primes via properties of L-functions and multiplicative functions.