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Some arithmetic convolution identities

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Abstract

Let n be a positive integer. Let \(\delta _3(n)\) denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic convolution sum

$$\begin{aligned} D_3(n):=\sum _{j=1}^{n-1}\delta _3(j)\delta _3(n-j) \end{aligned}$$

satisfies the relation

$$\begin{aligned} 3D_3(n)+\delta _3(n)={\sum _{\mathop {_{d \mid n}}\limits _{3 \not \mid d}}}d. \end{aligned}$$

In this paper, we use a result about binary quadratic forms to prove a general arithmetic convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new convolution formulas.

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Authors and Affiliations

  1. Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Kenneth S. Williams

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  1. Kenneth S. Williams
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Correspondence to Kenneth S. Williams.

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Williams, K.S. Some arithmetic convolution identities. Ramanujan J 43, 197–213 (2017). https://doi.org/10.1007/s11139-015-9745-1

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  • DOI: https://doi.org/10.1007/s11139-015-9745-1

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