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Jan. 2005 A generalization on the difference between an integer and its inverse modulo $q$. (II)
Tianping Zhang, Wenpeng Zhang
Proc. Japan Acad. Ser. A Math. Sci. 81(1): 7-11 (Jan. 2005). DOI: 10.3792/pjaa.81.7

## Abstract

Let $q > 2$ and $c$ are two integers with $(q, c) = 1$, for each integer $a$ with $0 < a < q$ and $(a, q) = 1$, there exists one and only one $b$ with $0 < b < q$ such that $ab \equiv c \pmod{q}$. Let $M(q,k,c,n) = \underset{a_1 \dotsm a_n b \equiv c \pmod{q}} {\sideset{}{'}\sum_{a_1=1}^q \dotsm \sideset{}{'}\sum_{a_n=1}^q \sideset{}{'}\sum_{b=1}^q} (a_1 \dotsm a_n - b)^{2k},$ the main purpose of this paper is to study the asymptotic behavior of $M(q,k,c,n)$, and prove that for any positive integers $k$ and $n$ with $n \ge 2$ we have $M(q,k,c,n) = \frac{\phi^n(q) q^{2kn}}{(2k+1)^n} + O \Bigl( 4^k q^{(2k+1)n - (1/2)} d^2(q) \ln q \Bigr).$

## Citation

Tianping Zhang. Wenpeng Zhang. "A generalization on the difference between an integer and its inverse modulo $q$. (II)." Proc. Japan Acad. Ser. A Math. Sci. 81 (1) 7 - 11, Jan. 2005. https://doi.org/10.3792/pjaa.81.7

## Information

Published: Jan. 2005
First available in Project Euclid: 18 May 2005

zbMATH: 1088.11072
MathSciNet: MR2068483
Digital Object Identifier: 10.3792/pjaa.81.7

Subjects:
Primary: 11F20 , 11N37

Keywords: Asymptotic formula , generalization  