On the Average Value of a Function of the Residual Index

Abstract

For a prime p and a positive integer a relatively prime to p, we denote $i_a(p)$ as the index of the subgroup generated by a in the multiplicative group $\left( \mathbb {Z}/p\mathbb {Z}\right) ^{\times }$. Under certain conditions on the arithmetic function f(n), we prove that the average value of $f(i_a(p))$, as a and p vary, is

$$\sum _{d=1}^{\infty } \frac{g(d)}{d\varphi (d)},$$

where $g(n)=\sum _{d\mid n} \mu (d) f(n/d)$ is the Möbius inverse of f and $\varphi (n)$ is the Euler function.

In honor of V. Kumar Murty on his sixtieth birthday

Research of the first author is partially supported by NSERC. Research of the second author is partially supported by a PIMS postdoctoral fellowship.