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.
Repository Citation
Pollack, Paul and Lola Thompson. 2015. "Arithmetic functions at consecutive shifted primes." International Journal of Number Theory 11(5): 1477-1498.
Publisher
World Scientific Publishing
Publication Date
8-1-2015
Publication Title
International Journal of Number Theory
Department
Mathematics
Document Type
Article
DOI
https://dx.doi.org/10.1142/S1793042115400023
Keywords
Maynard-Tao theorem, Shifted prime, Arithmetic functions, Sum of digits
Language
English
Format
text
