Arithmetic functions at consecutive shifted primes

2014/05/17 by Paul Pollack, Lola Thompson, Pollack, Paul +1 · 1 citation
Mathematics · #11N05 (Secondary) #11N37 (Primary) #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11N05 #msc:11N37

paper · pdf · doi:10.48550/arxiv.1405.4444

Made some improvements in the organization and exposition

openalex publication_date 2014/05/17 · arxiv created 2014/08/06 · arxiv updated 2014/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each of the functions f ∈ \ϕ, σ, ω, τ\ 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) < … < f(p_n+k-1), and similarly for f(p_n-1) > f(p_n+1-1) > … > f(p_n+k-1). We also answer some questions of Sierpiński 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.

Cited by

• Primes in intervals of bounded length

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• Some heuristics on the gaps between consecutive primes
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• Various Arithmetic Functions and their Applications
• Arithmetic progressions of primes in short intervals