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Evaluation of the convolution sums ∑ak + bl + cm = nσ(k)σ(l)σ(m) with lcm(a,b,c)≤6
- Title
- Evaluation of the convolution sums ∑ak + bl + cm = nσ(k)σ(l)σ(m) with lcm(a,b,c)≤6
- Authors
- Park Y.K.
- Ewha Authors
- 박윤경
- SCOPUS Author ID
- 박윤경
- Issue Date
- 2016
- Journal Title
- Journal of Number Theory
- ISSN
- 0022-314X
- Citation
- Journal of Number Theory vol. 168, pp. 257 - 275
- Keywords
- Convolution sum; Quasimodular form; The number of representation by quadratic forms
- Publisher
- Academic Press Inc.
- Indexed
- SCIE; SCOPUS
- Document Type
- Article
- Abstract
- The generating functions of divisor functions are quasimodular forms and their products belong to a space of quasimodular forms of higher weight. In this work, we evaluate the convolution sums ∑ak+bl+cm=nσ(k)σ(l)σ(m) for all positive integers a,b,c,n with lcm(a,b,c)≤6. The evaluation of this convolution sum in the case (a,b,c)=(1,1,1) is due to Lahiri [17] and in the cases (a,b,c)=(1,1,2),(1,2,2) and (1,2,4) to Alaca, Uygul and Williams [7]. As an application, the known formulas for the number of representations of a positive integer n by each of the quadratic forms∑j=012xj 2 and ∑j=16(x2j−1 2+x2j−1x2j+x2j 2) are reproved using new identities proved in this paper. © 2016 Elsevier Inc.
- DOI
- 10.1016/j.jnt.2016.04.025
- Appears in Collections:
- 연구기관 > 수리과학연구소 > Journal papers
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