The convergence of certain Diophantine series

[摘要] For x irrational, we study the convergence of series of the form Sigma n(-s)f (nx) where f is a real-valued, 1-periodic function which is continuous, except for singularities at the integers with a potential growth. We show that it is possible to fully characterize the convergence set and to approximate the series in terms of the continued fraction of x. This improves and generalizes recent results by Rivoal who studied the examples f(t) = cot(pi t) and f(t) = sin(-2)(pi t). (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Diophantine series;Lipschitz regularity;Approximate functional equation