Irregularities of Dirichlet L-Functions and a Parity Bias in Gaps of Zeros
报告专家：戈振超（加拿大滑铁卢大学）
报告时间：2024年6月25日（星期二）上午10：00-11：00
报告地点：国家天元数学西南中心505
报告摘要：The integral of Hardy's Z-function from $0$ to $T$ measures the occurrence of its sign changes. Hardy proved that this integral is $o(T)$ from which he deduced that the Riemann zeta-function has infinitely many zeros on the critical line. A. Ivić conjectured this integral is $O(T^{1/4})$ and $\Omega_{\pm}(T^{1/4})$ as $T\to\infty$. These estimates were proved, independently, by M. A. Korolev and M. Jutila.
In this talk, we will show that the analogous conjecture is false for the Z-functions of certain ``special" Dirichlet L-functions. In particular, we show that the integral of the Z-function of a Dirichlet L-functions from $0$ to $T$ is asymptotic to $c_\chi T^{3/4}$ and we classify precisely when the constant $c_\chi$ is nonzero. Experimentally, we find that the L-functions in this (thin) family have a significant and previously undetected bias in the distribution of gaps between their zeros. These phenomena appear to have an arithmetic explanation that corresponds to the non-vanishing of a certain Gauss- type sum.
This is joint work with Jonathan Bober and Micah Milinovich.
专家简介：戈振超，密西西比大学博士，现为滑铁卢大学Fields博士后。主要研究方向是解析数论和加性组合。 相关成果发表于Mathematische Zeitschrift, Acta Arithmetica 等著名期刊。
邀请人：胡京辰
