Title

Arithmetic functions at consecutive shifted primes

Abstract

For each of the functions f is an element of {phi, sigma, omega , tau} and every natural number K, we show that there are infinitely many solutions to the inequalities f(p(n) - 1) < f(p(n+1) - 1) < center dot center dot center dot < f(p(n+K) - 1), and similarly for f(p(n) - 1) > f(p(n+1) - 1) > center dot center dot center dot > f(p(n+ K) - 1). We also answer some questions of Sierpinski on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.

Publisher

World Scientific Publishing

Publication Date

8-1-2015

Publication Title

International Journal of Number Theory

Department

Mathematics

Document Type

Article

DOI

10.1142/S1793042115400023

Keywords

Maynard-Tao theorem, Shifted prime, Arithmetic functions, Sum of digits

Language

English

Format

text

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