G\'al-type GCD sums beyond the critical line

TL;DR

This paper establishes an optimal bound for GCD sums of Gál type for integers and exponents less than 1/2, revealing no influence of the Riemann zeta function's functional equation in this range.

Contribution

It provides a new optimal bound for GCD sums beyond the critical line, extending previous results and clarifying the absence of zeta functional equation effects for 0<α<1/2.

Findings

Proves an optimal bound for GCD sums for 0<α<1/2

Shows the bound is tight up to the exponent b(α)

Demonstrates no trace of the zeta functional equation in this range

Abstract

We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2\alpha}}{(n_k n_{\ell})^{\alpha}} \ll N^{2-2\alpha} (\log N)^{b(\alpha)} \] holds for arbitrary integers $1\le n_1<\cdots < n_N$ and $0<\alpha<1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(\alpha)$. This estimate complements recent results for $1/2\le \alpha \le 1$ and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<\alpha<1/2$.