A Tsang-range high-moment bound for $\operatorname{Im}\log L(\tfrac12+it,\chi)$ under GRH

Abstract

Conditional on the Generalized Riemann Hypothesis for $L(s,\chi)$, we prove the Selberg--Tsang high-moment bound for $X_\chi(t) = \operatorname{Im}\log L(\tfrac12+it,\chi)$ 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 $\log|\zeta(\tfrac12+it)|$.