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arxiv: 2512.01779 · v3 · pith:VSQPULI4new · submitted 2025-12-01 · 🧮 math.NT

A discrete approach to Dirichlet L-functions, their special values and zeros

Anders Karlsson , Dylan M\"uller This is my paper

Pith reviewed 2026-05-21 17:56 UTC · model grok-4.3

classification 🧮 math.NT
keywords Dirichlet L-functionsspecial valueszerosdiscrete spectral sumscyclic graphsEuler-Maclaurin expansiongeneralized Riemann hypothesisspanning forests
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The pith

Finite spectral sums on cyclic graphs turn asymptotic approximations of Dirichlet L-functions into exact identities at integer points.

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

The paper develops a discrete spectral framework that approximates Dirichlet L-functions by finite sums L_n(s, χ) linked to cyclic graphs. Pairing a refined Euler-Maclaurin expansion with a structural polynomiality property of these sums causes the asymptotic series to terminate exactly when the argument is an integer. This produces new infinite families of exact relations among special values of Dirichlet L-functions and recovers earlier formulas by a different route. The same termination also supplies a combinatorial reading of the special values as counts of rooted spanning forests on any fixed cyclic graph. For the zeros, the framework reformulates the generalized Riemann hypothesis for odd primitive characters as an asymptotic functional equation between the completed discrete functions ξ_n at s and 1-s.

Core claim

At integer arguments the asymptotic expansions of the finite spectral sums L_n(s, χ) terminate exactly due to their structural polynomiality, producing exact identities for special values of Dirichlet L-functions. These identities include new infinite families of relations and recover prior results by a different mechanism. The special values admit interpretations as counts related to rooted spanning forests on cyclic graphs. For the zeros, the framework reformulates the generalized Riemann hypothesis for odd primitive characters via an asymptotic functional equation linking the completed discrete functions ξ_n at s and 1-s.

What carries the argument

The asymptotic-to-exact principle, which combines a refined Euler-Maclaurin expansion with the structural polynomiality of the finite spectral sums L_n(s, χ) on cyclic graphs to terminate the expansion precisely at integer arguments.

If this is right

  • New infinite families of relations among special values of Dirichlet L-functions are obtained.
  • Formulas previously obtained by Xie, Zhao and Zhao are recovered through this discrete mechanism.
  • ζ(2n) and corresponding special values for all Dirichlet L-functions receive a finite combinatorial interpretation in terms of rooted spanning forests on any fixed cyclic graph.
  • The generalized Riemann hypothesis for odd primitive characters is reformulated as an asymptotic functional equation relating ξ_n(1-s, χ-bar) to ξ_n(s, χ) of the completed discrete functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The combinatorial forest interpretation may allow graph algorithms to compute or bound special values for large conductors.
  • Similar discrete approximations could be tested on other families of L-functions or on the Riemann zeta function itself.
  • The one-dimensional asymptotic functional equation picture completed here might suggest discrete models for testing GRH numerically on finite graphs.

Load-bearing premise

The structural polynomiality property of the finite spectral sums that makes the asymptotic expansion terminate exactly at integer arguments.

What would settle it

A concrete calculation for a fixed integer s, a specific character χ, and sufficiently large n showing that the terminated asymptotic expression from L_n(s, χ) fails to equal the known special value of L(s, χ).

read the original abstract

We develop a discrete spectral framework for Dirichlet $L$-functions that reveals a combinatorial structure underlying their special values and connects this to their zeros. Our approach approximates the classical Dirichlet series by finite spectral sums $L_n(s,\chi)$ associated with cyclic graphs $\mathbb{Z}/n\mathbb{Z}$ and studies their asymptotics as $n\rightarrow \infty$. Combining a refined Euler Maclaurin expansion with a structural polynomiality property, we show that at integer arguments the asymptotic expansions terminate and yield exact identities. This asymptotic to exact principle produces new infinite families of relations among special values of Dirichlet $L$-functions and recovers, by a different mechanism, formulas previously obtained by Xie, Zhao and Zhao. An interesting feature of our method is that $\zeta(2n)$ and the corresponding special values for all Dirichlet $L$-functions thereby admit a finite combinatorial interpretation in terms of rooted spanning forests on any fixed cyclic graph. Concerning zeros, the same framework leads to some remarks about real zeros and a reformulation of the Generalized Riemann Hypothesis in the case of odd primitive characters in terms of an asymptotic functional equation relating $\xi_n(1-s,\overline{\chi})$ to $\xi_n(s,\chi)$ of the completed discrete functions. This establishes the remaining case of the one dimensional picture obtained in earlier works.

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

1 major / 2 minor

Summary. The manuscript develops a discrete spectral framework for Dirichlet L-functions by approximating them via finite spectral sums L_n(s, χ) on cyclic graphs Z/nZ. Combining a refined Euler-Maclaurin expansion with a claimed structural polynomiality property of these sums at positive integer arguments, the asymptotics as n → ∞ are shown to terminate exactly, producing exact identities. This yields new infinite families of relations among special values, recovers formulas of Xie-Zhao-Zhao by a different route, supplies a combinatorial interpretation of ζ(2n) and L-values via rooted spanning forests on fixed cyclic graphs, and reformulates GRH for odd primitive characters as an asymptotic functional equation between completed discrete functions ξ_n.

Significance. If the polynomiality property is proved rigorously and the termination argument is free of remainders, the work supplies a genuinely new combinatorial route to special values of L-functions together with a discrete reformulation of GRH in the odd-primitive case. The finite spanning-forest expressions and the recovery of prior formulas are concrete strengths that would be of interest to the number-theory community.

major comments (1)
  1. [discussion of the asymptotic-to-exact principle and the polynomiality property] The structural polynomiality of L_n(s, χ) at positive integers s is the load-bearing assumption that converts the refined Euler-Maclaurin expansion into an exact identity. A self-contained proof (or at least a complete derivation) of this property for arbitrary n, s ∈ ℕ and general characters χ must be supplied; without it the claimed exact relations and the combinatorial interpretation via spanning forests do not follow.
minor comments (2)
  1. The notation for the completed discrete functions ξ_n(s, χ) should be introduced and fixed at the first appearance rather than after the functional-equation discussion.
  2. Precise citations to the specific theorems of Xie, Zhao and Zhao that are recovered should be added when the recovery is asserted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the constructive identification of the central assumption requiring further justification. We address the major comment point by point below.

read point-by-point responses

  1. Referee: The structural polynomiality of L_n(s, χ) at positive integers s is the load-bearing assumption that converts the refined Euler-Maclaurin expansion into an exact identity. A self-contained proof (or at least a complete derivation) of this property for arbitrary n, s ∈ ℕ and general characters χ must be supplied; without it the claimed exact relations and the combinatorial interpretation via spanning forests do not follow.

    Authors: We agree that the structural polynomiality property is the key step that turns the asymptotic expansion into an exact identity and that a self-contained derivation is required for the manuscript to be complete. In the revised version we will add a dedicated section (or appendix) containing a full derivation of this property. The argument proceeds by expressing the finite spectral sum L_n(s, χ) explicitly as a sum over roots of unity, applying the character orthogonality relations, and showing that the resulting expression is a polynomial in n of degree at most s-1 whose coefficients are independent of n for fixed s and χ. This derivation will be carried out uniformly for arbitrary positive integers n and s and for general Dirichlet characters χ (including the non-primitive case). With this addition the passage from the refined Euler-Maclaurin formula to the exact identities, as well as the combinatorial interpretation in terms of rooted spanning forests on the fixed cyclic graph, becomes fully rigorous. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via discrete spectral sums and Euler-Maclaurin termination

full rationale

The paper derives exact identities for special values by combining a refined Euler-Maclaurin expansion with the structural polynomiality of the finite sums L_n(s, χ) at positive integers, where the polynomiality is a property of the cyclic-graph spectral sums rather than a fitted parameter or self-referential definition. The resulting asymptotic-to-exact principle yields the claimed relations and combinatorial interpretations in terms of rooted spanning forests as consequences of the setup. The GRH reformulation is presented as an additional remark using the same framework and references earlier works only for contextual completion of a one-dimensional picture, without that citation serving as load-bearing justification for the central special-values claims. No equation or step reduces by construction to its own inputs, and the framework remains independent of external benchmarks or prior fitted results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework relies on a refined Euler-Maclaurin expansion and an unstated structural polynomiality property of the finite sums; no explicit free parameters, new entities, or ad-hoc axioms are named in the abstract, but the termination claim depends on these background analytic tools.

axioms (1)
  • domain assumption The finite spectral sums L_n(s, chi) possess a structural polynomiality property that makes the asymptotic expansion terminate exactly at integer s.
    Invoked when the abstract states that combining the refined Euler-Maclaurin expansion with this property yields exact identities.

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