Levinson's theorem and its generalization for Dirichlet L-functions
Swapnil Ray
TL;DR
This work presents a streamlined version of Levinson's mollified moment method, following Matthew P. Young's 2018 approach, to establish a concrete lower bound on the proportion of non-trivial zeros of the Riemann zeta function on the critical line. It then generalizes the method to Dirichlet L-functions, showing that more than $\tfrac{2}{5}$ of zeros lie on the critical line, with a positive proportion of these zeros being simple, by employing a longer mollifier and a carefully crafted Dirichlet polynomial mollifier. The paper derives explicit lower bounds $\kappa(\chi) > 0.4172$ and $\kappa^*(\chi) > 0.4074$ for primitive Dirichlet characters, under suitable growth conditions on $q$ and $T$, via asymptotics for the mollified second moment $I_R(Q,\chi)$ and an approximate functional equation. The results strengthen the understanding of zero-distribution for L-functions, connect to classical bounds (e.g., Hardy, Selberg, Levinson, Conrey), and contribute toward the broader conjecture that a positive, potentially large, share of zeros lie on the critical line with implications for the Generalized Riemann Hypothesis.
Abstract
In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) = 1/2, using a mollified second moment of the zeta-function. Later, we present a generalized result for Dirichlet L-functions by Xiaosheng Wu in 2018, using Levinson's method, showing that more than two-fifths of the non-trivial zeros of Dirichlet L-functions are on the critical line. Moreover, more than two-fifths of the non-trivial zeros are simple and on the critical line, using a longer mollifier than in Levinson's original proof. This generalizes a result by Conrey from 1989 that the Riemann zeta-function has at least two-fifths of its zeros on the critical line.