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

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 $$\frac{1}{y}\sum _{a\le y}\sum _{p\le x}\frac{1}{l_a(p)}=\log x+C\log \log x+O\left(\frac{x}{y\log \log x}\right)$$ 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 $$\frac{1}{y}\sum _{a<y}\sum _{\begin{array}{c}p<x\\ l_a(p)>\frac{x}{\psi (x)}\end{array}}1=\pi (x)+O\left(\frac{x\log x}{\psi (x)}\right)+O\left(\frac{x^{2-\delta }{\log }^{2}x}{y}\right).$$ Furthermore, if $x^{1-\delta }{\log }^{3}x=o(y)$, then $$\frac{1}{y}\sum _{a<y}\sum _{\begin{array}{c}p<x\\ p\nmid a\end{array}}{l}_a(p)=c\text{Li}(x^2)+O\left(\frac{x^2}{{\log }^{A}x}\right)+O\left(\frac{x^{3-\delta }{\log }^{2}x}{y}\right)$$ where $$c=\prod _p\left(1-\frac{p}{p^3-1}\right).$$

References in corpus (2)

• On the period of the linear congruential and power generators
• On the Order of $a$ modulo $n$ on Average

Cited by in corpus (1)

• Explicit constants in averages involving the multiplicative order