# A Variant of the Truncated Perron's Formula and Primitive Roots

## TL;DR

This paper demonstrates, assuming the Generalised Riemann Hypothesis, that most primes in a certain range have the expected number of primitive roots within specific intervals, using a novel variant of Perron's formula.

## Contribution

It introduces a new variant of the truncated Perron's formula to analyze prime primitive roots under GRH, extending previous results.

## Key findings

- Almost all primes in [Q, 2Q] have expected primitive roots in [x, x + x^{1/2 + δ}]
- The method applies for Q up to x^{2/3 - ε}
- Conditional on GRH, the distribution of primitive roots aligns with predictions

## Abstract

We show under the Generalised Riemann Hypothesis that for every $\delta>0$, almost every prime $q$ in $[Q,2Q]$ has the expected of prime primitive roots in the interval $[x,x+x^{\frac{1}2+\delta}]$ provided $Q$ is not more than $x^{\frac{2}{3}-\epsilon}$. We obtain this via a variant of the classical truncated Perron's formula for the partial sums of the coefficients of a Dirichlet series.