Mean value from representation of rational number as sum of two Egyptian fractions

For given positive integers n and a, let R(n;a) denote the number of positive integer solutions (x,y) of the Diophantine equationan=1x+1y. WriteS(N;a)=∑n⩽N(n,a)=1R(n;a). Recently Jingjing Huang and R.C. Vaughan proved that for 4⩽N and a⩽2N, there is an asymptotic formulaS(N;a)=3π2a∏p|ap−1p+1⋅N(log2N...

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Bibliographic Details
Published inJournal of number theory Vol. 132; no. 4; pp. 701 - 713
Main Author Jia, Chaohua
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2012
Subjects
Egyptian fraction
L-function
Mean value
L-function
Egyptian fraction
Mean value
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Summary:For given positive integers n and a, let R(n;a) denote the number of positive integer solutions (x,y) of the Diophantine equationan=1x+1y. WriteS(N;a)=∑n⩽N(n,a)=1R(n;a). Recently Jingjing Huang and R.C. Vaughan proved that for 4⩽N and a⩽2N, there is an asymptotic formulaS(N;a)=3π2a∏p|ap−1p+1⋅N(log2N+c1(a)logN+c0(a))+Δ(N;a). In this paper, we shall get a more explicit expression with better error term for c0(a).
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2011.09.007