Average Results on the Order of $a$ modulo $p$

TL;DR

This paper investigates the average behavior of the multiplicative order of integers modulo primes, providing improved asymptotic formulas and new average results under certain growth conditions.

Contribution

It offers new asymptotic formulas for the average order of $a$ modulo $p$, improving previous results and establishing multiple average estimates with explicit error bounds.

Findings

Derived an improved average formula for 1/l_a(p) over a and p.

Established new average results for large l_a(p) exceeding x/ψ(x).

Provided asymptotic estimates for the sum of l_a(p) over a and p.

Abstract

Let $a>1$ be an integer. Denote by $l_a(p)$ the multiplicative order of $a$ modulo primes $p$. We prove that if $\frac{x}{\log x\log\log x}=o(y)$, then $$\frac 1 y \sum_{a\leq y}\sum_{p\leq x}\frac{1}{l_a(p)}=\log x + C\log\log x+O\left(\frac x {y \log\log x}\right) $$ which is an improvement over a theorem by Felix ~\cite{Fe}. Additionally, we also prove two other average results If $\log^2 x = o(\psi(x))$ and $x^{1-\delta}\log^3 x = o(y)$, then $$\frac1y \sum_{a<y} \sum_{\substack{{p<x} \\ {l_a(p)>\frac{x}{\psi(x)}}}} 1 = \pi(x) + O\left(\frac{x\log x}{\psi(x)}\right) + O\left(\frac{x^{2 - \delta}\log^2 x}y\right).$$ Furthermore, if $x^{1-\delta}\log^3 x = o(y)$, then $$\frac1y\sum_{a<y} \sum_{\substack{{p<x} \\ {p\nmid a}}}l_a(p) = c\textrm{Li}(x^2) + O\left( \frac{x^2}{\log^A x} \right) + O\left(\frac{x^{3 -\delta}\log^2 x}y\right)$$ where $$c = \prod_p \left(1-\frac p{p^3-1}\right).$$