A note on Eisenstein series and convolution of sums

Abstract

The Eisenstein series plays a central role in modern number theory and mathematical analysis, especially in the theory of modular forms. Its applications span arithmetic, combinatorics, and mathematical physics. Eisenstein series play a significant role in mathematical physics, primarily due to their modular and automorphic properties. They arise naturally in string theory, where non-holomorphic Eisenstein series encode S-duality symmetries and appear in the coefficients of higher-order terms in superstring amplitudes. In quantum field theory and conformal field theory, Eisenstein series contribute to modular-invariant partition functions and describe lattice sums associated with compactified dimensions. In this paper, we investigate connections between Borweins' cubic theta functions and modular forms of level 6. Utilizing these relationships, we obtain an explicit representation of an Eisenstein series of level 6. In addition, we deduce several convolution sum identities of the form $\displaystyle{\sum_{2i+3j=m}\sigma(i)\sigma(j)}$, $\displaystyle{\sum_{i+6j=m}\sigma(i)\sigma(j)}$ and $\displaystyle{\sum_{i+8j=m}\sigma(i)\sigma(j)}$ which illustrate the interplay between theta functions and arithmetic functions.