Explicit constants in averages involving the multiplicative order

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 $p\le x$ on average. When we average over $a\le N$, 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 $y\ge \exp ((2+\epsilon )\sqrt{\log x\log \log x})$ in ~\cite[Theorem 1]{LP}, to $y>\exp (3.42\sqrt{\log x})$.

13 pages

References in corpus (1)

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