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

TL;DR

This paper proves an asymptotic formula for the average order of the sum of divisors function over shifted primes, revealing a logarithmic growth pattern in the sum as x becomes large.

Contribution

It provides the first proof of the asymptotic behavior of the sum of divisors over shifted primes, extending to any fixed integer shift.

Findings

Asymptotic formula for sum over primes involving sum of divisors

Growth rate of the sum is proportional to x log log x

Results hold for any fixed integer shift a

Abstract

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ \sum_{p\leq x}\sigma([x/p])=c_0x\log \log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ \sum_{p\leq x}\sigma([x/(p+a)])=c_0x\log \log x+O(x) $ for any fixed integer $a$.