S. D. Hughes

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math.NT2026

Few small primes suffice to cover unit groups

Scott D. Hughes

Klurman, Shparlinski and Teräväinen showed that there is a set of at most $(lo g Q)^{1 + ε}$ primes of polylogarithmic size whose subset products cover the unit group $(\ma…

math.NT2026

Effective logarithmic two-point Chowla bounds in every window

Scott D. Hughes

We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window. For L_i(n)=a_i n+b_i with…

math.NT2026

The density of sums of distinct divisors

Scott D. Hughes

For a positive integer $t$ , let $d_{t}$ denote the natural density of the set of $n$ for which $t$ is a sum of distinct divisors of $n$ . Erdős proved that $d_{t}$ exists, gave an unspe…

math.NT2026

A Tsang-range high-moment bound for $Im lo g L (\frac{1}{2} + i t, χ)$ under GRH

Scott D. Hughes

Conditional on the Generalized Riemann Hypothesis for $L (s, χ)$ , we prove the Selberg--Tsang high-moment bound for $X_{χ} (t) = Im lo g L (\frac{1}{2} + i t, χ)$ at fixed squaref…

Few small primes suffice to cover unit groups

Effective logarithmic two-point Chowla bounds in every window

The density of sums of distinct divisors

A Tsang-range high-moment bound for ImlogL(21​+it,χ) under GRH

A Tsang-range high-moment bound for $Im lo g L (\frac{1}{2} + i t, χ)$ under GRH