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

TL;DR

This paper derives an asymptotic formula for a sum involving the Euler totient function over shifted primes, revealing new insights into prime-related number theoretic functions.

Contribution

It provides the first asymptotic evaluation of a sum involving the Euler totient of the largest integer function over primes.

Findings

Asymptotic formula for the sum over primes involving ([x/p]) and (n)

Identification of main terms including rac{6}{pi^2}x rac{log log x}{log x}

Establishment of an error term of order O(x/ log x)

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.