Explicit approaches to L-functions and primes

黎曼猜想是数学中最受追捧的猜想之一。它在素数分布中的应用是基本的。黎曼ζ函数的零点与素数之间的关系是解析数论领域的起源。该计划的哲学在于优化和创建分析数论工具，以产生与数字相关的完全明确的结果。这种策略的突出例子可以在Helfgott关于哥德巴赫猜想的工作中找到。完全描述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。黎曼假设及其推广版本断言黎曼ζ函数（以及其他更一般的l函数族）的所有复非平凡零点都位于穿过1 / 2的垂直线上。我们对质数的理解主要取决于经过1的垂直线的左边有多远的地方有0，以及有多少个0。我计划证明新的零密度结果，以研究关于连续幂之间质数的问题。与zeta函数不同，l函数具有“低”零（虚部较小），并且可能接近1（称为“例外”）。这些零实际上在理解质数（在等差数列中，在数字字段中）方面起着重要作用。一个有用的信息是，可能的例外零对其他零有排斥作用。我建议研究零的位置和密度。该方案包括增大无零区域和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。结合各种平滑参数，筛边界和数值计算，我的目标是改进以前对各种有限和和乘积的显式估计。许多用来研究Dirichlet l -函数的零点和等差数列中的素数的工具可以推广到Hecke l -函数和Chebotarev密度定理。特别地，我将探索Chebotarev密度定理中最小素数和误差项的大小。我对Lang-Trotter猜想的应用和素数检验的界也很感兴趣。

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.