Prime numbers and L-functions

数学中研究最多的对象之一是黎曼ζ函数。1859年，著名的德国数学家伯恩哈德·黎曼写了一篇关于数论的文章。在这本重要的著作中，他提出质数可以通过黎曼ζ函数来研究。此外，他还提供了一种通过这个函数来研究质数的具体方法。这种方法涉及到研究某些数字，这些数字被称为黎曼ζ函数的零点。黎曼提出了一个关于零的简单而深刻的猜想。黎曼假设是黎曼猜想所有的零都在一条直线上。这仍然是数学中最重要的未解决问题之一。它激发了一些在不同数学领域工作的最优秀的研究人员进行大量的研究。它的分辨率将为我们目前对质数的认识提供更精细的信息。这个猜想的推广，广义黎曼假设，也被认为是正确的。这个猜想涉及到l函数零点的位置。l函数是黎曼函数的变体。我的研究涉及l函数的行为和它们的零点。这些函数还编码有关素数的深层信息。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好的行为。综上所述，我提出通过涉及素数和乘法函数的工具来研究l函数，另一方面，我将通过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.