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

Abstract

Let $x\ge 1$ 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 ${\sum }_{p\le x}\sigma ([x/p])=c_{0}x\log \log x+O(x)$ over the primes, where $c_0>0$ is a constant. More generally, ${\sum }_{p\le x}\sigma ([x/(p+a)])=c_{0}x\log \log x+O(x)$ for any fixed integer $a$.

Nine Pages. Keywords: Arithmetic function; Sum of divisors function; Largest integer function; Average orders; Shifted primes. arXiv admin note: substantial text overlap with arXiv:2105.00790

References in corpus (1)

• Average Orders of the Euler Phi Function, The Dedekind Psi Function, The Sum of Divisors Function, And The Largest Integer Function