Arithmetic functions at consecutive shifted primes

TL;DR

This paper proves the existence of infinitely many consecutive shifted primes with ordered values of various arithmetic functions and addresses questions on digit sums of consecutive primes, using advanced prime distribution methods.

Contribution

It establishes new infinite patterns in shifted primes for functions like , , , , and answers Sierpi
4dnski's questions on digit sums, employing Maynard and Tao's prime interval techniques.

Findings

Infinite solutions for inequalities involving , , ,  at consecutive shifted primes.

Confirmed existence of ordered and reversed sequences of function values at consecutive shifted primes.

Addressed Sierpi
4dnski's questions on digit sums of consecutive primes.

Abstract

For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1}-1) > \dots > f(p_{n+k}-1)$. We also answer some questions of Sierpi\'nski 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.