Explicit evaluation of triple convolution sums of the divisor functions

IF 0.5 3区数学 Q3 MATHEMATICS

International Journal of Number Theory Pub Date : 2024-04-27 DOI:10.1142/s1793042124500544

B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh

引用次数: 0

Abstract

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums $W_{d_{1}, d_{2}, d_{3}}^{r_{1}, r_{2}, r_{3}} (n) = \sum_{\binom{l_{1}, l_{2}, l_{3} \in ℕ}{d_{1} l_{1} + d_{2} l_{2} + d_{3} l_{3} = n}} σ_{r_{1}} (l_{1}) σ_{r_{2}} (l_{2}) σ_{r_{3}} (l_{3}),$ for odd integers $r_{1}, r_{2}, r_{3} \geq 1,$ and $d_{1}, d_{2}, d_{3}, n \in ℕ$ , where $σ_{r} (n)$ is the sum of the $r$ th powers of the positive divisors of $n$ . We consider four cases, namely (i) $r_{1} = r_{2} = r_{3} = 1$ , (ii) $r_{1} = r_{2} = 1$ ; $r_{3} \geq 3$ (iii) $r_{1} = 1$ ; $r_{2}, r_{3} \geq 3$ and (iv) $r_{1}, r_{2}, r_{3} \geq 3$ , and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer $n$ by certain positive definite quadratic forms. The existing formulas for $W_{1, 1, 1} (n)$ (in [20]), $W_{1, 1, 2} (n), W_{1, 2, 2} (n), W_{1, 2, 4} (n)$ (in [7]), $W_{1, 1, 1}^{1, 3, 3} (n), W_{1, 1, 3}^{1, 3, 3} (n), W_{1, 3, 3}^{1, 3, 3} (n), W_{3, 1, 1}^{1, 3, 3} (n), W_{3, 3, 1}^{1, 3, 3} (n)$ (in [35]), $W_{d_{1}, d_{2}, d_{3}} (n)$ , $lcm (d_{1}, d_{2}, d_{3}) \leq 6$ (in [30]) and $lcm (d_{1}, d_{2}, d_{3}) = 7, 8, 9$ (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

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除数函数三重卷积和的显式计算

本文利用模形式理论，给出了求卷积和 Wd1,d2,d3r1,r2,r3(n)=∑l1,l2 的一般方法、l3∈ℕd1l1+d2l2+d3l3=nσr1(l1)σr2(l2)σr3(l3)，对于奇整数 r1,r2,r3≥1，以及 d1,d2,d3,n∈ℕ，其中 σr(n) 是 n 的正除数的 r 次幂和。我们考虑了四种情况，即 (i) r1=r2=r3=1，(ii) r1=r2=1; r3≥3 (iii) r1=1; r2,r3≥3 和 (iv) r1,r2,r3≥3，并给出了各自卷积和的明确表达式。我们举例说明了每种情况下的卷积和，并进一步利用这些公式得到了某些正定二次型对正整数 n 的表示数的明确公式。现有公式为 W1,1,1(n) (见 [20]), W1,1,2(n), W1,2,2(n), W1,2,4(n) (见 [7]), W1,1,11,3,3(n), W1,1,31,3,3(n), W1,3,31,3,3(n), W3,1,11,3,3(n), W3,3,11,3,3(n) (见 [35])、Wd1,d2,d3(n),lcm(d1,d2,d3)≤6（在 [30] 中）和 lcm(d1,d2,d3)=7,8,9 （在 [31] 中），都是利用准模态理论得到的。

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来源期刊

International Journal of Number Theory 数学-数学

CiteScore

1.10

自引率

14.30%

发文量

审稿时长

4-8 weeks

期刊介绍： This journal publishes original research papers and review articles on all areas of Number Theory, including elementary number theory, analytic number theory, algebraic number theory, arithmetic algebraic geometry, geometry of numbers, diophantine equations, diophantine approximation, transcendental number theory, probabilistic number theory, modular forms, multiplicative number theory, additive number theory, partitions, and computational number theory.