This paper is devoted to a weighted version of the one-level density of the non-trivial zeros of $L$-functions, tilted by a power of the $L$-function evaluated at the central point. Assuming the Riemann Hypothesis and the ratio conjecture, for some specific families of $L$-functions we prove that the same structure suggested by the density conjecture holds also in this weighted investigation, if the exponent of the weight is small enough. Moreover we speculate about the general case, conjecturing explicit formulae for the weighted kernels.
Moments in the Chebotarev density theorem: non-Gaussian families
Université Paris Cité, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche（巴黎西岱大学，索邦大学，法国国家科学研究中心，朱西厄-巴黎左岸数学研究所）；Univ. Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学，法国国家科学研究中心，奥赛数学实验室）；Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学，法国国家科学研究中心，波尔多国立高等电子与计算机学院，IMB）
One-level density of zeros of the Riemann zeta function weighted by its special values
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University（金泽大学理工学部）；Faculty of Mathematics, Kyushu University（九州大学数学系）
Homological spectral graph theory and weighted cycle counting
School of Informatics, Xiamen University（厦门大学信息学院）；Department of Mathematics and Statistics, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）
