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Evaluation of the convolution sums ∑ak + bl + cm = nσ(k)σ(l)σ(m) with lcm(a,b,c)≤6
- Evaluation of the convolution sums ∑ak + bl + cm = nσ(k)σ(l)σ(m) with lcm(a,b,c)≤6
- Park Y.K.
- Ewha Authors
- SCOPUS Author ID
- Issue Date
- Journal Title
- Journal of Number Theory
- vol. 168, pp. 257 - 275
- Convolution sum; Quasimodular form; The number of representation by quadratic forms
- Academic Press Inc.
- SCI; SCIE; SCOPUS
- 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  and in the cases (a,b,c)=(1,1,2),(1,2,2) and (1,2,4) to Alaca, Uygul and Williams . 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.
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