On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

Authors

  • Mehdi Hassani Department of Mathematics University of Zanjan University Blvd., 45371-38791, Zanjan, Iran
  • Mahmoud Marie Department of Mathematics University of Zanjan University Blvd., 45371-38791, Zanjan, Iran

DOI:

https://doi.org/10.37256/cm.152020546

Keywords:

Euler function, residue systems, arithmetic function, alternating sum

Abstract

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑ runs through all integers k with 1 ≤ kn and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function, we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑nx φm(n) and ∑nxφG(n), the alternative averages ∑nx(1)n−1φm(n) and ∑nx(−1)n−1φG(n).

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Published

2020-11-05

How to Cite

1.
Hassani M, Marie M. On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments. Contemp. Math. [Internet]. 2020 Nov. 5 [cited 2024 Apr. 18];1(5):417-22. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/546