A Tsang-range high-moment bound for $\mathrm{Im}\log L(\frac{1}{2}+it,\chi )$ under GRH

Abstract

Conditional on the Generalized Riemann Hypothesis for $L(s,\chi )$, we prove the Selberg--Tsang high-moment bound for $X_{\chi }(t)=\mathrm{Im}\log L(\frac{1}{2}+it,\chi )$ at fixed squarefree odd conductor $q\ge 3$ and primitive non-principal character $\chi$. Writing $L_T=\log \log (qT)$: for every $K>0$ there exist constants $C_K$ and $T_0$ such that $\frac{1}{T}{\int }_T^{2T}|X_{\chi }(t){|}^{2k}dt\le (C_{K}k{L}_T{)}^k$ for all $T\ge T_0$ and every integer $1\le k\le K{L}_T$. The proof ports Selberg's pointwise approximate formula for $S(t)$ to $L(s,\chi )$ at fixed conductor under GRH, splits it into three prime-power Dirichlet polynomials, and evaluates their moments via Soundararajan's mean-value lemma. As a corollary, Markov's inequality yields a Gaussian-scale tail $\exp (-c{V}^2/L_T)$ for $\sqrt{L_T}\ll V\ll L_T$ -- a GRH-conditional, fixed-conductor, imaginary-part analogue of the large-deviation upper bounds known for $\log |\zeta (\frac{1}{2}+it)|$.