Let n be a positive integer. Let (Formula presented.) 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 (Formula presented.)satisfies the relation (Formula presented.)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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Keywords Arithmetic convolution identities, Divisor functions, Representations by binary quadratic forms, Sums of two binary quadratic forms
Persistent URL dx.doi.org/10.1007/s11139-015-9745-1
Journal Ramanujan Journal
Williams, K.S. (2017). Some arithmetic convolution identities. Ramanujan Journal, 1–17. doi:10.1007/s11139-015-9745-1