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We are motivated by Ramanujan’s recursion formula for sums of the product of two Eisenstein series (Berndt in Ramanujan’s Notebook, Part II, 1989, Entry 14, p.332) and its proof, and also by Besge-Liouville’s convolution identity for the ordinary divisor function σk−1(n)Open image in new window (Williams in Number Theory in the Spirit of Liouville, vol. 76, 2011, Theorem 12.3). The objective of this paper is to introduce and prove convolution identities for the twisted divisor functions σk−1∗(n)Open image in new window as well as for the twisted Eisenstein series S2k+2,χ0Open image in new window and S2k+2,χ1Open image in new window, S2k+2∗Open image in new window, S2k+2,χ0∗Open image in new window, and S2k+2,χ1∗Open image in new window. As applications based on our main results, we establish many interesting identities for pyramidal, triangular, Mersenne, and perfect numbers. Moreover, we show how our main results can be used to obtain arithmetical formulas for the number of representations of an integer n as the sums of s squares.

Convolution identities for twisted Eisenstein series and twisted divisor functions