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A mean-value theorem on sums of two k-th powers of numbers in residue classes

A mean-value theorem on sums of two k-th powers of numbers in residue classes Abh. Math. Sem. Univ. Hamburg 60 (1990), 249-256 A Mean-Value Theorem on Sums of Two k-th Powers of Numbers in Residue Classes By G. KUBA 1 Introduction For positive integers k > 2 and n, let rk(n) denote the number of ways to write n as a sum of two k-th powers of natural numbers. To study the "average order" of this arithmetic function, one considers Dirichlet's summatory function Rk(x) = ~ rk(n), n~<x where x is a large real variable. Of course, the evaluation of Rk(X) is a problem of lattice point theory in the classical sense of Landau. (See the recent textbook of KRJiTZEL [2] for an enlightening survey of the field.) It has been proved by KR~,TZEL [3] that 2(1 oo I'~(~) X~ --X ~ +ClX~-~ ' y'n-l-~ 9 , ~ sm(2znx~ - ~) + o(x=) Rk(X) = 2kF(2) n=l with an effective constant cl depending on k. Consequently, for k > 3, r2(~) x~ - x~ + O(x~-b) Rk(x) = 2kF@ and r2(~) x~ - x~ + n_+(x~-~). Rk(x) = 2kF(2) (For k = 2, this is the famous circle problem of Gauss; see again [2] for its history; of course, the term involving the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

A mean-value theorem on sums of two k-th powers of numbers in residue classes

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02941060
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Abstract

Abh. Math. Sem. Univ. Hamburg 60 (1990), 249-256 A Mean-Value Theorem on Sums of Two k-th Powers of Numbers in Residue Classes By G. KUBA 1 Introduction For positive integers k > 2 and n, let rk(n) denote the number of ways to write n as a sum of two k-th powers of natural numbers. To study the "average order" of this arithmetic function, one considers Dirichlet's summatory function Rk(x) = ~ rk(n), n~<x where x is a large real variable. Of course, the evaluation of Rk(X) is a problem of lattice point theory in the classical sense of Landau. (See the recent textbook of KRJiTZEL [2] for an enlightening survey of the field.) It has been proved by KR~,TZEL [3] that 2(1 oo I'~(~) X~ --X ~ +ClX~-~ ' y'n-l-~ 9 , ~ sm(2znx~ - ~) + o(x=) Rk(X) = 2kF(2) n=l with an effective constant cl depending on k. Consequently, for k > 3, r2(~) x~ - x~ + O(x~-b) Rk(x) = 2kF@ and r2(~) x~ - x~ + n_+(x~-~). Rk(x) = 2kF(2) (For k = 2, this is the famous circle problem of Gauss; see again [2] for its history; of course, the term involving the

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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Aug 28, 2008

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