Explicit approaches to L-functions and primes

黎曼猜想是数学中最受欢迎的猜想之一。它在素数分布上的应用是基本的。黎曼zeta函数的零点和素数之间的关系是解析数论领域的起源。该计划的理念包括优化和创建解析数论工具，以产生完全明确的结果，这是数值相关的。这种策略的突出例子可以在赫尔夫戈特关于哥德巴赫猜想的工作中找到。完全描述性的结果也可以直接应用于其他数学领域，如组合学或密码学。黎曼猜想及其推广版本断言，黎曼zeta函数（以及其他更一般的L-函数族）的所有复数非平凡零点都沿着通过1/2的垂直线。我们对素数的理解基本上依赖于零在通过1的垂直线的左边有多远，以及有多少个零。我计划证明新的零密度结果，以研究连续幂之间的素数问题。与zeta函数不同的是，L函数有“低位”的零（虚部很小），可能还有一个接近1的零（称为“例外”）。这些零实际上在理解素数（在算术级数中，在数域中）中起着重要作用。一个有用的信息是，可能的例外零对其他零有排斥作用。我打算研究零点的位置和密度。该方案包括扩大无零区域和对Dirichlet和Hecke L-函数具有更强的排斥现象。我还想展示1线附近零的稀缺性。我将应用这些结果来探索几个素数定理。再加上各种平滑参数，筛界和数值计算，我的目标是改善以前的显式估计各种有限和产品的素数。许多研究算术级数中Dirichlet L-函数和素数零点的工具可以推广到Hecke L-函数和Chebotarev密度定理。特别是我会探讨的大小最小的素数和错误的条款在Chebotarev密度定理。我也感兴趣的应用程序的朗-特罗特猜想和界素测试。

The Riemann Hypothesis is one of the most sought after conjectures in mathematics. Its application to the distribution of prime numbers is fundamental. The relation between zeros of the Riemann zeta function and primes is at the origin of the field of analytic number theory. The philosophy of this program consists in optimizing and creating analytic number theory tools in order to produce completely explicit results which are numerically relevant. Prominent examples of this strategy can be found in the work of Helfgott on the Goldbach conjecture. Fully descriptive results can also then be directly applied to other fields of mathematics like combinatorics or cryptography. The Riemann Hypothesis and its generalized versions assert that all complex non-trivial zeros of the Riemann zeta function (and other more general families of L-functions) sit along the vertical line passing through ½. Our understanding of primes relies essentially on how far left from the vertical line passing through 1 the zeros are located and how many of them there are. I plan to prove new zero-density results in order to investigate questions about primes between consecutive powers. Unlike the zeta function, L-functions have ``low-lying" zeros (with small imaginary part), and possibly one close to 1 (referred to as "exceptional"). These zeros actually play a significant role in understanding primes (in arithmetic progressions, in number fields). A useful information is the fact that the possible exceptional zero has a repulsion effect on other zeros. I propose to investigate the location and density of zeros. This program includes enlarging zero-free regions and having stronger repulsion phenomenon for Dirichlet and Hecke L-functions. I also want to exhibit scarcity of zeros near the 1-line. I would apply these results to explore several prime number theorems. Together with various smoothing arguments, sieve bounds, and numerical computations, I aim to improve previous explicit estimates for various finite sums and product over prime numbers. Many of the tools developed to study zeros of Dirichlet L-functions and primes in arithmetic progressions can be generalized to Hecke L-functions and to the context of Chebotarev density theorem. In particular I would explore the size of the least prime and of error terms in the Chebotarev density theorem. I am also interested in applications to the Lang-Trotter conjecture and to bounds for primality testing.