On the correlation of the sum of digits along prime numbers
Volume 199 / 2021
Acta Arithmetica 199 (2021), 275-301
MSC: Primary 11A07, 11A41, 11A63, 11L07, 11L20; Secondary 11B50, 11L03, 11N13.
DOI: 10.4064/aa200514-22-12
Published online: 19 April 2021
Abstract
Let $s_{q}$ denote the sum of digits function in base $q$. The aim of this work is to estimate the exponential sums involving the sum of digits of shifted prime numbers, of the form $\sum _{p\leq x}e(\alpha _{0}s_{q}(p)+\cdots +\alpha _{k}s_{q}(p+k))$, where $k$ is a positive integer and $\alpha _{0},\ldots ,\alpha _{k}$ are real numbers. We deduce from these estimates some results concerning the correlation along prime numbers of the Thue–Morse sequence.
