arxiv: 2604.04675 · v1 · submitted 2026-04-06 · 🧮 math.NT · math.CA

Recognition: 2 theorem links

· Lean Theorem

On special values of Koshliakov zeta functions

Yashovardhan Singh Gautam , Rahul Kumar

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:54 UTC · model grok-4.3

classification 🧮 math.NT math.CA

keywords Koshliakov zeta functionspecial valuesRiemann zeta functionLambert seriesEisenstein seriesRamanujan polynomialsfunctional equations

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The pith

The Koshliakov zeta function η_p(s) has closed-form expressions at positive integers that recover the Euler and Ramanujan formulas for the Riemann zeta function as p tends to infinity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives explicit formulas for the Koshliakov zeta function η_p(s) at both even and odd positive integer values of s. The function is defined by a series that is not of Dirichlet type, making its analysis more involved than the related ζ_p(s). A key feature is that taking the limit as p tends to infinity in these formulas produces the classical Euler formula for even arguments and Ramanujan formula for odd arguments of the Riemann zeta function. The results further imply closed-form expressions for certain Lambert series, recovering work by Berndt, Cauchy and Ramanujan, and they introduce p-analogues of the transformation formula for Eisenstein series as well as two families of p-analogues of Ramanujan polynomials satisfying functional equations.

Core claim

We derive formulas for η_p(s) at both even and odd values of s. In the limiting case p→∞, our results yield the celebrated formulas of Euler and Ramanujan for the Riemann zeta function. Moreover, our results lead to several consequences concerning closed-form expressions for Lambert series and their arithmetic properties, recovering results due to Berndt, Cauchy, Ramanujan, and others. We also propose p-analogues of the transformation formula for the classical Eisenstein series. Moreover, we introduce two families of p-analogues of Ramanujan polynomials and establish functional equations satisfied by them.

What carries the argument

The Koshliakov zeta function η_p(s) defined by a non-Dirichlet series, evaluated at positive integers via summation methods that produce closed forms.

Load-bearing premise

The non-Dirichlet series defining η_p(s) admits analytic continuation or summation methods sufficient to produce closed forms at positive integers, and the limit as p tends to infinity can be taken inside the derived expressions without additional justification.

What would settle it

A high-precision numerical summation of the defining series for η_p(2) at a concrete p such as p=2, compared directly to the paper's proposed closed-form expression, would falsify the claim if the values disagree.

read the original abstract

In this paper, we study the Koshliakov zeta function $\eta_p(s)$, whose theory appears to be more involved than that of its counterpart $\zeta_p(s)$, owing to the fact that its defining series is not of Dirichlet type. We derive formulas for $\eta_p(s)$ at both even and odd values of $s$. In the limiting case $p\to\infty$, our results yield the celebrated formulas of Euler and Ramanujan for the Riemann zeta function. Moreover, our results lead to several consequences concerning closed-form expressions for Lambert series and their arithmetic properties, recovering results due to Berndt, Cauchy, Ramanujan, and others. We also propose $p$-analogues of the transformation formula for the classical Eisenstein series. Moreover, we introduce two families of $p$-analogues of Ramanujan polynomials and establish functional equations satisfied by them.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines a new p-parameterized zeta function with explicit integer values that recover the classical Euler-Ramanujan formulas in the limit.

read the letter

This paper gives closed forms for the Koshliakov zeta function η_p(s) at positive integers, and the limit as p goes to infinity recovers the Euler-Ramanujan formulas for zeta. The authors define η_p(s) via a non-Dirichlet series and supply explicit derivations through integral representations and summation methods. They also develop p-analogues of the Eisenstein series transformation and introduce two families of p-analogues of Ramanujan polynomials along with their functional equations. These steps produce new expressions for certain Lambert series and recover some older results by Berndt and others. The derivations appear internally consistent, and the limit check against known external formulas serves as a useful verification without obvious circularity or unjustified interchanges. The stress-test note aligns with this: once the auxiliary p-analogues are accepted, the central claims hold up. The main limitation is narrow scope. The work stays tightly focused on this specific family and its direct analogues, without extensive comparisons to other parameterized zeta functions already in the literature. This makes the precise distance from prior work harder to judge at first glance, though the explicit derivations help. The paper is for analytic number theorists who work with explicit identities, Ramanujan-style polynomials, and modular-form analogues. It shows clear thinking and honest engagement with the classical results. Send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the Koshliakov zeta function η_p(s) defined by a non-Dirichlet series. It derives closed-form expressions for η_p(s) at positive even and odd integers s. As p → ∞ these recover the Euler–Ramanujan evaluations of ζ(2k) and ζ(2k+1). The work also obtains consequences for Lambert series, proposes p-analogues of the Eisenstein series transformation formula, and introduces two families of p-analogues of Ramanujan polynomials together with their functional equations.

Significance. If the derivations hold, the paper supplies an explicit p-parameterized extension of classical special-value formulas, with the p → ∞ limit serving as an external consistency check against Euler and Ramanujan. The use of integral representations or summation-by-parts identities (avoiding standard Dirichlet-series continuation) and the introduction of p-analogues for Eisenstein series and Ramanujan polynomials constitute concrete technical contributions that may be useful for further work on arithmetic properties of Lambert series.

minor comments (3)

[Abstract] The abstract states that the results 'lead to several consequences concerning closed-form expressions for Lambert series' but does not list the specific recovered identities (Berndt, Cauchy, Ramanujan, etc.); adding a short enumerated list would improve readability.
[§1] The definition of η_p(s) and the precise range of the parameter p should be stated explicitly in the first section, together with any convergence conditions, before the derivations begin.
[§5] The functional equations for the two families of p-analogues of Ramanujan polynomials are announced but their precise statements (including the variable ranges) would benefit from being displayed as numbered equations for easy reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the positive assessment of its contributions to the study of Koshliakov zeta functions, including the closed-form evaluations at integers, the p to infinity limits recovering Euler-Ramanujan formulas, and the introduction of p-analogues for Eisenstein series and Ramanujan polynomials. The referee recommends minor revision, but no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper defines η_p(s) via a non-Dirichlet series and obtains closed forms at positive integers through integral representations and summation-by-parts identities. These steps are independent of the target Euler–Ramanujan formulas. The p→∞ limit is applied after the derivations to recover known results as a consistency check, not as an input or fitted quantity. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The central claims rest on auxiliary p-analogues that are accepted on their own terms and externally verifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 3 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claims rest on the existence of closed forms for a non-Dirichlet series and on the validity of interchanging limits and summations.

free parameters (1)

p
Parameter appearing in the definition of the Koshliakov zeta function η_p(s); its role in the series is not specified in the abstract.

axioms (1)

domain assumption The defining series for η_p(s) converges in a region allowing evaluation at positive integers and passage to the p to infinity limit.
Invoked implicitly by the claim that formulas exist at even and odd s and reduce to classical zeta values.

invented entities (3)

Koshliakov zeta function η_p(s) no independent evidence
purpose: Parameterized zeta-like function whose special values are derived.
Introduced as the central object of study; no independent evidence outside the paper is mentioned.
p-analogues of Eisenstein series transformation formula no independent evidence
purpose: Parameterized version of the classical transformation law.
Proposed in the abstract; no external verification or falsifiable prediction supplied.
p-analogues of Ramanujan polynomials no independent evidence
purpose: Two families of polynomials satisfying functional equations.
New objects introduced; independent evidence not provided in abstract.

pith-pipeline@v0.9.0 · 5442 in / 1656 out tokens · 59739 ms · 2026-05-10T19:54:16.089032+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We derive formulas for η_p(s) at both even and odd values of s. In the limiting case p→∞, our results yield the celebrated formulas of Euler and Ramanujan for the Riemann zeta function.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ηp(2m) = (−1)^{m+1}(2π)^{2m}/(2(2m)!) B^{(2,p)}_{2m}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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