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An extension of Ramanujan-Guinand identity for the Dedekind zeta function and a new formula for $ζ_\mathbb{F}(2)$ and $ζ_\mathbb{F}(2m+1)$

This paper establishes a number field analogue of the Ramanujan-Guinand identity for the Dedekind zeta function, from which the authors derive new and elegant formulas for $\zeta_\mathbb{F}(2)$ and $\zeta_\mathbb{F}(2m+1)$ for any number field $\mathbb{F}$ .

Original authors: Diksha Rani Bansal, Bibekananda Maji

Published 2026-08-10

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CC BY 4.0

Original authors: Diksha Rani Bansal, Bibekananda Maji

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe of numbers as a vast, cosmic library where every book is a number, and the stories they tell are hidden in patterns. For over a century, mathematicians have been trying to decode the most mysterious stories in this library: the "special values" of the Riemann zeta function. Think of this function as a magical machine that takes a number as an input and spits out a result that reveals deep secrets about how prime numbers are distributed. We know exactly what this machine does when you feed it even numbers (like 2, 4, 6), but when you feed it odd numbers (like 3, 5, 7), the machine goes silent, and the answers remain stubbornly hidden. This is one of the biggest unsolved puzzles in mathematics.

To make sense of this, mathematicians often look for "mirror images" or "transformation formulas." Imagine a kaleidoscope: if you turn it one way, you see a complex pattern; turn it the other way, and you see a different pattern that is mathematically identical to the first. In the 1920s, the legendary mathematician Srinivasa Ramanujan discovered a stunning identity—a perfect mirror relationship—that connected these mysterious odd-number values to a rapidly converging series (a sum that gets to the answer very quickly). This was later rediscovered and formalized as the Ramanujan-Guinand identity. It's like finding a secret code that translates a difficult problem into an easy one. But this code was written for the standard number system. What if we wanted to use it in more exotic, complex number worlds? That is the question this paper tackles.

In this new study, authors Diksha Rani Bansal and Bibekananda Maji take Ramanujan's famous mirror code and expand it to work in "number fields." You can think of a number field as a different country in the land of mathematics, with its own unique rules, landscapes, and inhabitants (ideals and integers). While the standard number system is like a flat, familiar plain, these number fields are like rugged, multi-dimensional terrains. The authors successfully build a bridge, creating a "number field analogue" of the Ramanujan-Guinand identity. They prove that the same beautiful mirror relationship holds true even in these complex, exotic lands.

The paper's main finding is a powerful new formula that acts as a universal translator for these number fields. By establishing this new identity, the authors achieve two major feats. First, they recover the classic Ramanujan-Guinand identity and a related formula by Ramanujan-Koshliakov as special cases, proving their new theory is consistent with the old masters' work. Second, and perhaps more excitingly, they use this new tool to derive a brand-new formula for calculating $\zeta_F(2)$ (the value of the zeta function at 2) for any number field. Previously, a mathematician named Zagier had found a way to calculate this, but his formula involved a tricky logarithmic integral. The authors' new formula, however, expresses the answer as a rapidly converging series, which is often much easier to compute and understand.

Furthermore, the team derives an elegant identity for $\zeta_F(2m + 1)$ (values at odd integers) for any positive integer $m$ . This is significant because, as mentioned earlier, odd values are notoriously difficult to pin down. The authors show that by using their new mirror formula, they can express these elusive values in terms of other rapidly converging series. They didn't just guess; they provided a rigorous mathematical proof, using tools like the Dedekind zeta function (the version of the zeta function for these exotic number fields) and a special function called the Steen function (which acts like a sophisticated filter to handle the complex geometry of these fields).

To demonstrate that their new formula works in the real world, the authors applied it to several specific quadratic number fields (like $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{5})$ ). They calculated the values of $\zeta_F(2)$ for these fields and presented the results in a table, showing that their method yields precise numerical answers. This confirms that their theoretical bridge is sturdy enough to carry calculations across the most complex mathematical terrains. In short, this paper doesn't just solve a puzzle; it builds a new, more versatile tool that allows mathematicians to see the hidden patterns of numbers in a whole new light, turning the silent, mysterious odd values of the zeta function into a chorus of rapidly converging, calculable series.

Technical Summary: An Extension of Ramanujan-Guinand Identity for the Dedekind Zeta Function

Problem Statement
The Riemann zeta function $\zeta(s)$ and its special values have long been central to analytic number theory. While Euler provided closed-form evaluations for even integers $\zeta(2m)$ , no such simple formula exists for odd integers $\zeta(2m+1)$ . Ramanujan discovered a remarkable identity (the Ramanujan-Guinand identity) relating a generalized divisor function and a modified Bessel function, which connects to the transformation of Eisenstein series and yields formulas for odd zeta values.

While analogues of Ramanujan's identity for odd zeta values have been extended to number fields, a comprehensive number field analogue of the Ramanujan-Guinand identity itself—specifically for the Dedekind zeta function $\zeta_F(s)$ —remained to be established. Furthermore, while Zagier (1986) provided a formula for $\zeta_F(2)$ involving logarithmic integrals, a rapidly convergent series formula for $\zeta_F(2)$ and $\zeta_F(2m+1)$ for arbitrary number fields was an open area of exploration.

Methodology
The authors, Diksha Rani Bansal and Bibekananda Maji, employ the theory of automorphic forms and complex analysis to derive their results. The core methodology involves:

Fourier Expansion of Eisenstein Series: The authors utilize the Fourier expansion of the non-holomorphic Eisenstein series $G(\tau, s)$ over the full modular group, which naturally incorporates the divisor function and modified Bessel functions.
Functional Equations: They leverage the functional equation of the Dedekind zeta function $\zeta_F(s)$ and the product $\zeta_F(s)\zeta_F(s+z)$ .
Steen Function and Meijer G-Functions: To generalize the modified Bessel functions appearing in the classical identity to arbitrary number fields, the authors introduce the Steen function $V(x|a_1, \dots, a_n)$ , defined via a Mellin-Barnes integral. This function is a specific case of the Meijer G-function, allowing for the handling of the varying degrees of real ( $r_1$ ) and complex ( $r_2$ ) embeddings in a number field $F$ .
Contour Integration: The proofs rely on shifting contours in the complex plane and applying Cauchy's residue theorem to the product of gamma factors and zeta functions, specifically analyzing the poles at $s=0, 1, -z,$ and $1-z$ .
Mellin Transform Techniques: The authors use Mellin transforms to relate the series involving the Steen function to the product of zeta functions, establishing the transformation formulas.

Key Contributions and Results

Main Theorem (Theorem 2.1): The paper establishes a number field analogue of the Ramanujan-Guinand identity. For a number field $F$ of degree $d$ , the authors prove the transformation formula:
$I_{F,z}(x) = I_{F,-z}\left(\frac{1}{x}\right)$
where $I_{F,z}(x)$ involves a series of generalized divisor functions $\sigma_{F,-z}(n)$ weighted by the Steen function $V_{F,z}$ , plus a term involving the leading coefficient of the Laurent expansion of $\zeta_F(s)$ at $s=0$ and the functional equation factor $\Omega_F(z)$ .
Recovery of Classical Identities:
- Corollary 2.3: When $F = \mathbb{Q}$ , the main theorem reduces exactly to the classical Ramanujan-Guinand identity.
- Corollary 2.5: Setting $z=0$ and $F=\mathbb{Q}$ recovers the Ramanujan-Koshliakov identity.
- Corollary 2.7: Setting $z=1$ and $F=\mathbb{Q}$ yields a known identity of Ramanujan equivalent to the transformation formula for the logarithm of the Dedekind eta function.
New Formulas for Special Values:
- $\zeta_F(2)$ (Theorem 2.8): As a direct application of the $z=1$ case, the authors derive a new formula for $\zeta_F(2)$ for any arbitrary number field $F$ . Unlike Zagier's formula, which expresses $\zeta_F(2)$ via a logarithmic integral, this new formula presents $\zeta_F(2)$ as a rapidly convergent series involving the Steen function. The authors provide numerical evaluations for quadratic fields (e.g., $\mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{5})$ ) to demonstrate the formula's utility.
- $\zeta_F(2m+1)$ (Theorem 2.9): The authors derive an elegant identity for $\zeta_F(2m+1)$ for any positive integer $m$ . This formula expresses the odd zeta values in terms of rapidly convergent series involving the Steen function and $\zeta_F(2m+2)$ . Remark 4 highlights that by choosing $x$ as a primitive root of unity, one can isolate $\zeta_F(2m+1)$ explicitly.
Functional Equation of Products: Theorem 2.2 demonstrates that the transformation formula (2.1) implies the functional equation for the product $\zeta_F(s)\zeta_F(s+z)$ , reinforcing the deep connection between transformation formulas and functional equations.

Significance and Claims
The paper claims to provide a "number field analogue" of the Ramanujan-Guinand identity, thereby unifying the study of special values of zeta functions across different fields. The authors emphasize that their work illustrates the philosophy that transformation formulas for automorphic forms encode special values of zeta and L-functions.

Specifically, the significance lies in:

Generalization: Extending a classical identity from $\mathbb{Q}$ to arbitrary number fields $F$ .
Novelty in Evaluation: Providing a new, rapidly convergent series formula for $\zeta_F(2)$ that complements Zagier's integral-based evaluation.
Unification: Showing that the Ramanujan-Koshliakov identity and the eta function transformation are special cases of a broader number field identity.
Explicit Formulas: Offering explicit, computable identities for $\zeta_F(2m+1)$ , a class of values for which simple closed forms are generally unknown.

The authors conclude that their results offer another instance of the interplay between modular transformation formulas and the explicit evaluation of special values, providing tools for further investigation into the arithmetic of number fields.

Technical Summary: An Extension of Ramanujan-Guinand Identity for the Dedekind Zeta Function

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