Scott D. Hughes | Searcharxiv

arXiv2026

Sums of distinct divisors of factorials

For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le(2\log2+o(1))\,n/\log n$. This improves the bounds of order $n/(\log n)^{1/2-\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above $\sqrt{n!}$, with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant $2\log2$.

math.NT↗

arXiv2026

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

Conditional on the Generalized Riemann Hypothesis for $L(s,χ)$, we prove the Selberg--Tsang high-moment bound for $X_χ(t) = \operatorname{Im}\log L(\tfrac12+it,χ)$ at fixed squarefree odd conductor $q \ge 3$ and primitive non-principal character $χ$. Writing $L_T = \log\log(qT)$: for every $K > 0$ there exist constants $C_K$ and $T_0$ such that $\frac{1}{T}\int_T^{2T} |X_χ(t)|^{2k}\,dt \le (C_K\,k\,L_T)^k$ 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 $L(s,χ)$ 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 $\log|ζ(\tfrac12+it)|$.

math.NT↗