Ramanujan-type identities for alternating Hurwitz zeta functions

Abstract: Around 1910, in an unpublished manuscript, Ramanujan proposed the following identity for $ζ(2n+1)$: \begin{align*} α^{-n}\left{\frac{1}{2}ζ\left(2n+1\right) +\sum_{m=1}^{\infty}\frac{m^{-2n-1}}{e^{2αm}-1}\right} &-\left(-β\right)^{-n}\left{\frac{1}{2}ζ\left(2n+1\right) +\sum_{m=1}^{\infty}\frac{m^{-2n-1}}{e^{2βm}-1}\right} \&=2^{2n}\sum_{k=0}^{n+1}{\frac{\left(-1\right)^{k-1}B_{2k}B_{2n-2k+2}} {\left(2k\right)!\left(2n-2k+2\right)!}α^{n-k+1}β^k}, \end{align*} where $α$, $β$ are positive numbers satisfying $αβ=π^{2,n\in\mathbb} Z\setminus{0},$ $B_n$ denotes the $n$-th Bernoulli number, and $ζ(z)$ is the Riemann zeta function. In this paper, we extend Ramanujan's identity to the alternating Hurwitz zeta function and systematically investigate the properties of the alternating Hurwitz zeta function $ζ_E(z,x)$ under different modular symmetry conditions, as well as the corresponding Ramanujan-type identities. We also establish infinite series expressions for products of the tangent and hyperbolic tangent functions, and express the Dirichlet lambda function $λ(z)$ together with linear combinations of infinite series as convolution sums of special sequences. Furthermore, we define alternating Hurwitz kernels of even and odd orders, and obtain Ramanujan-type identities involving the alternating digamma function $\widetildeψ(x)$ and Euler polynomials $E_n(x)$, as well as transformation formulas between even-order and odd-order alternating Hurwitz kernels.

• The paper introduces new modular convolution identities and alternating kernels that extend Ramanujan-type formulas to alternating Hurwitz zeta functions.
• It employs advanced analytic techniques, including Mellin transforms and residue calculus, to derive explicit series involving Euler polynomials, Bernoulli numbers, and the Dirichlet lambda function.
• The results provide practical analytic tools and modular transformations with potential applications in number theory and the numerical evaluation of special zeta values.

Ramanujan-Type Identities for Alternating Hurwitz Zeta Functions

Introduction and Context

This work provides a comprehensive extension of classical Ramanujan-type identities for the Riemann zeta function to the framework of the alternating Hurwitz zeta function, $\zeta_E(s,x)$. The motivating classical result is Ramanujan's (circa 1910) modular transformation formula for $\zeta(2k+1)$, later formalized and generalized by Berndt, which connects analytic number theory, modular forms, and special values of zeta and $L$-functions. The present paper systematizes these connections in the alternating Hurwitz context, introducing new kernel functions, modular symmetry, and deriving explicit convolution sums for special functions including Euler polynomials, Bernoulli numbers, and the Dirichlet lambda function.

The Alternating Hurwitz Zeta Function and Kernels

The alternating Hurwitz zeta function is defined by

$$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$

and generalizes both the Dirichlet eta function and the Hurwitz zeta function. Notably, $\zeta_E(s,x)$ is entire in $s$, circumventing the pole at $s=1$ present for the standard Hurwitz zeta function, a property which is systematically exploited. The associated alternating digamma and gamma functions, $\widetilde{\psi}(x)$ and $\widetilde{\Gamma}(x)$, play key roles as analytic tools, and their asymptotics, recursion, and reflection properties are established and used throughout.

The authors define alternating Hurwitz kernels of both even and odd order:

• Even-order: $F(x,a;k)$ via an inverse Mellin-type integral with a cosine kernel and a corresponding series involving $\zeta(2k+1)$0.
• Odd-order: $\zeta(2k+1)$1 replacing the cosine by a sine in the kernel, with a series involving $\zeta(2k+1)$2.

Such kernels generalize the Ramanujan kernel introduced by Chavan in the non-alternating setting, and satisfy modular-type transformations under $\zeta(2k+1)$3 (or variants thereof), which are essential for the main results.

Main Theorems and Results

Ramanujan-Type Convolution Formulas

The central results are modular-parameterized convolution formulas for $\zeta(2k+1)$4 and its products, generalizing those for $\zeta(2k+1)$5 and $\zeta(2k+1)$6:

• Even-Order Convolution Formula for $\zeta(2k+1)$7: For $\zeta(2k+1)$8, $\zeta(2k+1)$9 (or $L$0), the authors establish

$L$1

expressed as a sum over series involving the alternating digamma function $L$2, with modular covariance and explicit dependence on the ratio $L$3 and vice versa.

• Odd-Order Convolution for $L$4: For $L$5, a relation is developed for linear combinations of $L$6 and series involving $L$7, generating matched convolution sums on the right.
• Kernel Convolution Identities: Analogous to Chavan’s Hurwitz kernel identities, but generalized to the alternating setting and both even and odd order, with explicit double sums and series forms presented and modular symmetry made manifest.

Transformation and Explicit Series Identities

• Tangent/Hyperbolic Tangent Product Series: Under $L$8, the product $L$9 is represented as a convergent rational series indexed by odd integers—a generalization whose coefficients (for any fixed $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$0) admit analytic computation.
• Dirichlet Lambda Function Identity: A broad modular family of identities relates linear combinations involving $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$1 and infinite series to finite convolution sums over Bernoulli numbers, Euler polynomials at $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$2, and Genocchi numbers, extending Euler’s and Ramanujan’s classical evaluations for odd zeta values.
• Ramanujan-Type Identities Involving Euler Polynomials: For the alternating Hurwitz kernels, convolution identities with explicit Euler polynomial coefficients are proved, giving rise to modular relations for alternating zeta values at even/odd arguments.
• Transformation Formulas between Even/Odd Kernels: The paper also derives identities relating the even and odd kernel families, generalizing Ramanujan-type modular relationships between different parity zeta values.

Analytic and Algebraic Tools

The work extensively exploits complex analysis (contour integration and residue calculus), Mellin-Barnes integral transforms, and the analytic continuation properties of $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$3. Modular symmetry (specifically, for $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$4 or appropriate rational multiples) is central and is consistently used to obtain functional equations by symmetry and to close contour integrals.

The definitions and properties of $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$5 and $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$6 are given in detail, including asymptotics, reflection, and recursion, underscoring the technical depth of the functional analytic groundwork required.

The authors' development and systematic study of alternating invariant functions (following Zhu and Hu) connects the analytic results to an emerging algebraic theory, further showing closure under natural operations.

Implications

The results provide a detailed extension of the spectral and algebraic theory surrounding Ramanujan sums to the class of alternating Dirichlet series and their Hurwitz analogues. By connecting to Bernoulli, Euler, and Genocchi numbers, explicit modular convolution identities are made available not only for theoretical analysis but also for numerical computation of special values. The explicit infinite product, reflection, and recursion relations for $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$7 and $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$8 offer new analytic tools for the study of $$\zeta_E(s,x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+x)^s}, \qquad \Re(s) > 0, \; x \notin \mathbb{Z}_{\le0},$$9-functions, Eisenstein series, and their generalizations.

The modular transformation techniques and explicit convolution kernels offer new avenues for studying special values of alternating $\zeta_E(s,x)$0-functions, Eulerian-type series, and related invariants in cyclotomic fields, with potential connections to Stark’s conjectures and number theory at large.

The methodology—comprising inverse Mellin transforms and modular symmetries—suggests further generalization potential to non-commutative Dirichlet series, multiple zeta values, and $\zeta_E(s,x)$1-deformations. In particular, the interplay with Eisenstein series, Eichler integrals, and Lambert series is fertile ground for future research, especially in explicit evaluations and modular transformation properties.

Conclusion

This work establishes a comprehensive Ramanujan-type theory in the alternating Hurwitz zeta setting, providing both deep analytic foundation and explicit constructive identities for $\zeta_E(s,x)$2, kernel functions, and associated special values. The explicit convolution and series representations with modular parameters unify and extend much previous work on classical and Hurwitz zeta values, providing not only new theoretical insight but concrete analytic tools for further number-theoretic investigation.