Sum of Divisors Function And The Largest Integer Function Over The Shifted Primes

2021/06/30 by N. A. Carella, Carella, N. A.
Mathematics · #Analytic Number Theory Research #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.2107.01030

Abstract

Let x≥ 1 be a large number, let [x]=x-\x\ be the largest integer function, and let σ(n) be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order ∑_{p≤ x}σ([x/p])=c₀xlog log x+O(x) over the primes, where c₀>0 is a constant. More generally, ∑_{p≤ x}σ([x/(p+a)])=c₀xlog log x+O(x) for any fixed integer a.

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