Euler Totient 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 $\phi (n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes ${\sum }_{p\le x}\phi ([x/p])=(6/{\pi }^2)x\log \log x+c_{0}x+O\left(x(\log x{)}^{-1}\right)$, where $c_0$ is a constant, is evaluated in this note.

Nine Pages. Keywords: Shifted prime; Multiplicative function; Euler phi function; Average orders; Largest integer function