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

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 $a$.