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) \u3c f(p(n+1) - 1) \u3c center dot center dot center dot \u3c f(p(n+K) - 1), and similarly for f(p(n) - 1) \u3e f(p(n+1) - 1) \u3e center dot center dot center dot \u3e 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\u27s method for producing many primes in intervals of bounded length