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arxiv: 2604.02803 · v1 · submitted 2026-04-03 · 🧮 math.NT

Equivalence between the Functional Equation and Vorono\"{i}-type summation identities for a class of L-Functions

Arindam Roy , Jagannath Sahoo , Akshaa Vatwani This is my paper

Pith reviewed 2026-05-13 18:53 UTC · model grok-4.3

classification 🧮 math.NT
keywords L-functionsfunctional equationVoronoi summationDirichlet seriesarithmetic functionssummation identities
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The pith

The functional equation for a class of L-functions is equivalent to Voronoi-type summation identities for their arithmetic coefficients.

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

The paper proves that for Dirichlet series satisfying functional equations with multiple Gamma factors, an arithmetic function serves as the coefficient sequence if and only if it obeys the corresponding Voronoi-type summation formulas. The authors begin with the known implication from the functional equation to the summation identity and then establish the converse by showing that the summation identities force the required structural properties, including the functional equation itself. A sympathetic reader would care because this equivalence offers a characterization of such L-functions through summation properties that can be checked directly on the coefficients rather than through analytic continuation alone.

Core claim

By considering Dirichlet series satisfying functional equations involving multiple Gamma factors, we show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the Voronoi-type summation formulas. This equivalence is obtained by using the functional equation to derive the summation identity and conversely demonstrating that the summation identities yield the functional equation.

What carries the argument

The Voronoi-type summation identity for sums of the arithmetic coefficients against test functions, which is shown to be equivalent to the functional equation with multiple Gamma factors.

Load-bearing premise

The Dirichlet series are assumed to satisfy functional equations involving multiple Gamma factors.

What would settle it

An arithmetic function that satisfies the summation formulas yet produces a Dirichlet series whose analytic continuation fails to satisfy the expected functional equation with multiple Gamma factors.

read the original abstract

To date, the best methods for estimating the growth of mean values of arithmetic functions rely on the Vorono\"{\i} summation formula. By noticing a general pattern in the proof of his summation formula, Vorono\"{\i} postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ``nice" test functions $f(n)$, provided $a(n)$ is an ``arithmetic function". These arithmetic functions $a(n)$ are called so because they are expected to appear as coefficients of some $L$-functions satisfying certain properties. It has been well-known that the functional equation for a general $L$-function can be used to derive a Vorono\"{\i}-type summation identity for that $L$-function. In this article, we show that such a Vorono\"{\i}-type summation identity in fact endows the $L$-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish an if-and-only-if equivalence for a class of Dirichlet series whose completed L-functions involve an arbitrary but fixed number of Gamma factors: the coefficients a(n) appear in such a series if and only if they satisfy Voronoi-type summation identities for nice test functions. The forward direction (functional equation implies summation formula) is described as standard; the converse is the novel contribution.

Significance. If the equivalence is rigorously proved, it supplies a characterization of L-functions via their summation formulas rather than analytic continuation, which could streamline certain mean-value estimates for arithmetic functions. The result is of moderate interest in analytic number theory, particularly for work that already relies on Voronoi formulas, but its impact depends on whether the converse recovers the exact Gamma factors and conductor without additional data.

major comments (2)
  1. [Converse implication (around the statement of the main theorem)] The converse direction (summation identities imply the functional equation) requires showing that the identities uniquely determine the precise Gamma factors, their shifts, and the conductor. The manuscript does not appear to supply an explicit uniqueness argument via Mellin inversion against a sufficiently rich test-function class; without this, other meromorphic continuations could satisfy the same summation relations.
  2. [Definition of the class and statement of equivalence] The setup for the class of L-functions (Dirichlet series completed with multiple Gamma factors) is used to define both sides of the equivalence. It is not clear from the argument whether the recovery of the functional equation from the summation formulas is independent of the a-priori choice of Gamma factors or whether the test-function class separates the possible Gamma kernels.
minor comments (2)
  1. [Notation and setup] Clarify the precise Schwartz-class or Schwartz-type conditions imposed on the test functions f in the summation identities, including any decay or support requirements needed for the integral transforms.
  2. [Introduction] Add a brief comparison with existing uniqueness results for functional equations recovered from approximate functional equations or summation formulas in the literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will incorporate clarifications into the revised version.

read point-by-point responses

  1. Referee: The converse direction (summation identities imply the functional equation) requires showing that the identities uniquely determine the precise Gamma factors, their shifts, and the conductor. The manuscript does not appear to supply an explicit uniqueness argument via Mellin inversion against a sufficiently rich test-function class; without this, other meromorphic continuations could satisfy the same summation relations.

    Authors: We appreciate this observation. In the proof of the converse, the summation identities are applied to a dense class of test functions (smooth, compactly supported functions whose Mellin transforms are invertible). This allows direct recovery of the completed L-function via Mellin inversion, yielding the functional equation for the fixed Gamma factors, shifts, and conductor that define the class. We will add an explicit paragraph after the statement of the main theorem detailing this uniqueness step and confirming that no other meromorphic continuation satisfies the same identities for the given kernel. revision: yes

  2. Referee: The setup for the class of L-functions (Dirichlet series completed with multiple Gamma factors) is used to define both sides of the equivalence. It is not clear from the argument whether the recovery of the functional equation from the summation formulas is independent of the a-priori choice of Gamma factors or whether the test-function class separates the possible Gamma kernels.

    Authors: The class is defined with a fixed number of Gamma factors together with fixed shifts and conductor; the Voronoi-type summation formulas are stated with respect to the explicit integral kernel determined by those parameters. The converse therefore recovers the functional equation precisely for this a-priori choice. The test-function class is chosen so that its Mellin transforms separate distinct kernels. We will revise the statement of the main theorem and add a clarifying sentence in the introduction to emphasize that the equivalence holds for the given fixed Gamma data. revision: partial

Circularity Check

0 steps flagged

No circularity: equivalence derived from independent structural properties of the L-function class

full rationale

The paper establishes an if-and-only-if between arithmetic functions satisfying Voronoi-type summation identities and those serving as coefficients of Dirichlet series whose completed L-functions obey functional equations with multiple Gamma factors. The forward implication is described as standard (derivable from the functional equation), while the converse proceeds by showing that the summation identities endow the series with the required analytic continuation and functional equation. No self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain appears; the class is defined externally via the Gamma-factor functional equation, and the equivalence is proved within that fixed setup without collapsing back to the input identities by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard framework of Dirichlet series and their functional equations with Gamma factors; no new free parameters or entities are introduced.

axioms (1)
  • domain assumption Dirichlet series satisfy functional equations involving multiple Gamma factors
    This defines the specific class of L-functions for which the equivalence is claimed.

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