Euler Totient Function And The Largest Integer Function Over The Shifted Primes

Abstract

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq x}\varphi([x/p])=(6/\pi^2)x\log \log x+c_0x+O\left (x(\log x)^{-1}\right) $, where $c_0$ is a constant, is evaluated in this note.