Generalized Cotangent Series and Links Zeta and Theta Functions
Mahipal Gurram
TL;DR
The paper addresses generalizing cotangent-type series to higher-order lattice sums by defining $U_n(z)=\sum_{k\in\mathbb{Z}} \frac{1}{k^n+z^n}$ and introducing a generalized Jacobi theta function $\Psi_n(q)=\sum_{k\in\mathbb{Z}} q^{k^{2n}}$. It achieves a finite closed-form meromorphic representation for $U_n(z)$, proves a zeta-value limit $\zeta(2n)=\frac{1}{2}\lim_{z\to 0}(U_{2n}(z)-z^{-2n})$, a recursion for powers of two, and an integral representation $U_{2n}(z)=\int_0^1 q^{z^{2n}-1}\,\Psi_n(q)\,dq$ that ties to generalized theta kernels. The framework recovers classical cases $U_1(z)=\pi\cot(\pi z)$ and $U_2(z)=\frac{\pi}{z}\coth(\pi z)$ and unifies Ramanujan-type series with modular-analytic structures, suggesting broader connections to modular forms and Eisenstein-type sums. This work deepens the link between lattice sums, zeta values, and theta-type transformations, with potential implications for spectral sums in mathematical physics and number theory.
Abstract
This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine trigonometric and hyperbolic structures, uncover recursive patterns, and establish integral connections to generalized Jacobi theta functions. The analysis reveals deep structural relationships between lattice summations, values of the Riemann zeta function, and modular kernels of theta type. Using techniques such as contour integration, Mellin transforms, and factorization identities, we obtain new expressions for even zeta values and formulate integral identities linking these series to generalized theta functions. These results unify classical expansions of trigonometric and hyperbolic cotangent functions within a broader analytic framework, offering new insights into modular forms.