Explicit constants in averages involving the multiplicative order

TL;DR

This paper provides explicit constants and improved bounds for the average behavior of the multiplicative order of a modulo primes, refining previous asymptotic results and extending the applicable ranges.

Contribution

It explicitly determines constants in the asymptotic estimates of the average multiplicative order sums and extends the range of parameters where these estimates hold.

Findings

Explicit constants for the range of N in Stephens' theorem.

Improved lower bounds for the parameter y in related estimates.

Refined asymptotic estimates for the sum involving multiplicative orders.

Abstract

Let $a>1$. Denote by $l_a(p)$ the multiplicative order of $a$ modulo $p$. We look for an estimate of sum of $\frac{l_a(p)}{p-1}$ over primes $p\leq x$ on average. When we average over $a\leq N$, we observe a statistic of $C\mathrm{Li}(x)$. P. J. Stephens ~\cite[Theorem 1]{S} proved this statistic for $N>\exp(c_1\sqrt{\log x})$ for some positive constant $c_1$. Upon this result, we give an explicit value of $c_1$. In fact, ~\cite[Theorem 1, 3]{S} hold with $N>\exp(3.42\sqrt{\log x})$, and ~\cite[Theorem 2, 4]{S} hold with $N>\exp(4.8365\sqrt{\log x})$. Also, we improve the range of $y$, from $y\geq \exp((2+\epsilon)\sqrt{\log x\log\log x})$ in ~\cite[Theorem 1]{LP}, to $y > \exp(3.42\sqrt{\log x})$.