# On the difference in values of the Euler totient function near prime arguments

## TL;DR

This paper proves that for each fixed offset, the difference in Euler totient values at points near primes is equally likely to be positive or negative, demonstrating a balanced distribution.

## Contribution

It establishes an unconditional result on the equal distribution of the sign of the difference in totient values near primes for all fixed offsets.

## Key findings

- The difference in totient values near primes is positive for 50% of odd primes.
- The difference in totient values near primes is negative for 50% of odd primes.
- The result holds unconditionally for all fixed offsets ll  1.

## Abstract

We prove unconditionally that for each $\ell \geq 1$, the difference $\phi(p-\ell) - \phi(p+\ell)$ is positive for $50\%$ of odd primes $p$ and negative for $50\%$.