On the average value of a function of the residual index. (English) Zbl 1475.11171
Akbary, Amir (ed.) et al., Geometry, algebra, number theory, and their information technology applications. Toronto, Canada, June, 2016, and Kozhikode, India, August, 2016. Cham: Springer. Springer Proc. Math. Stat. 251, 19-37 (2018).
Summary: For a prime pand a positive integer arelatively prime to p, we denote \(i_a(p)\) as the index of the subgroup generated by ain the multiplicative group \((\mathbb{Z}/p\mathbb{Z})^{\times}\). Under certain conditions on the arithmetic function \(f(n)\), we prove that the average value of \(f(i_a(p))\), as aand pvary, is
\[
\sum\limits_{d=1}^{\infty}\frac{g(d)}{d\varphi (d)},
\]
where \(g(n)=\sum_{d\mid n}\mu (d)f(n/d)\) is the Möbius inverse of fand \(\varphi (n)\) is the Euler function.
For the entire collection see [Zbl 1403.11002].
For the entire collection see [Zbl 1403.11002].
MSC:
| 11N37 | Asymptotic results on arithmetic functions |
| 11A07 | Congruences; primitive roots; residue systems |
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