Gál-type GCD sums beyond the critical line

Abstract

We prove that N k,ℓ=1holds for arbitrary integers 1 ≤ n 1 < · · · < n N and 0 < α < 1/2 and show by an example that this bound is optimal, up to the precise value of the exponent b(α). This estimate complements recent results for 1/2 ≤ α ≤ 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 < α < 1/2.

Sommes de Gál et applications

“…Gál a résolu la conjecture d'Erdős correspondante dans le cas . De nombreux articles récents [, , ] concernent le comportement asymptotique de la quantité elles‐même liée aux majorations de certains polynômes de Dirichlet et à celles de maximums localisés de la fonction zêta de Riemann sur la droite verticale d'abscisse .…”

Decoupling theorems for the Duffin-Schaeffer problem

“…The case α ∈ (0, 1/2) seems to be much less natural. In this range the connection with the Riemann zeta function breaks down [8]. Similarly, the connection with sums of dilated functions breaks down, since the corresponding function would not be in L 2 anymore.…”

“…Let S r 1 k 1 and S r 2 k 2 be two sets of indices as above. Assume that both S r 1 k 1 and S r 2 k 2 are so large that they contribute to the divergence of the series (8). Then the set systems E m , m ∈ S r 1 k 1 and E n , n ∈ S r 2 k 2 are essentially independent, provided that either the elements of S r 1 k 1 and S r 2 k 2 , or the individual measures assigned to these elements, are of significantly different order.…”

Extreme values of derivatives of the Riemann zeta function

“…for some positive constants α(σ), β(σ)? (These bounds are inspired by the work of Bondarenko-Hilberdink-Seip in [9], where the authors studied GCD sums for σ ∈ (0, 1 2 ) ).…”