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arxiv: 2604.11941 · v1 · submitted 2026-04-13 · 🧮 math.NT

Simultaneous non-vanishing of Dirichlet L-functions

Hung M. Bui , Alexandra Florea , Micah B. Milinovich This is my paper

Pith reviewed 2026-05-10 15:11 UTC · model grok-4.3

classification 🧮 math.NT
keywords Dirichlet L-functionsnon-vanishingcritical linegeneralized Riemann hypothesisDirichlet characterssimultaneous non-vanishingtwisted L-functions
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The pith

Under GRH, a positive proportion of Dirichlet characters modulo large primes keep four twisted L-functions non-vanishing at any fixed critical-line point.

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

The paper establishes simultaneous non-vanishing for four Dirichlet L-functions at a fixed point 1/2 + it on the critical line. Fix even characters χ_j modulo pairwise coprime square-free D_j and fix t. Assuming the generalized Riemann hypothesis, when q is a prime large enough relative to the D_j and t, a positive proportion of characters χ modulo q satisfy L(1/2 + it, χ χ_j) ≠ 0 for every j = 1 to 4. Without GRH the authors still obtain infinitely many such χ modulo each large q, although the proportion shrinks to zero as q grows.

Core claim

Assuming the Generalized Riemann Hypothesis, the product ∏_{j=1}^4 L(1/2 + it, χ χ_j) is nonzero for a positive proportion of Dirichlet characters χ modulo q, where q is a sufficiently large prime depending on the fixed even characters χ_j mod D_j (pairwise coprime and square-free) and on t.

What carries the argument

The product of four twisted L-functions ∏ L(1/2 + it, χ χ_j) whose simultaneous non-vanishing is established for many χ mod q under GRH.

If this is right

  • The non-vanishing holds simultaneously across all four L-functions at the chosen point.
  • The result applies to any fixed real t and any fixed set of even characters χ_j with the stated conditions on the D_j.
  • The prime modulus q must exceed an explicit bound depending on the D_j and t.
  • Unconditionally, infinitely many characters χ mod q still satisfy the simultaneous non-vanishing, though their density tends to zero.

Where Pith is reading between the lines

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

  • The method may extend to families of L-functions attached to other arithmetic objects once analogous GRH assumptions are available.
  • The explicit dependence of the size of q on D_j and t makes the result potentially checkable by direct computation for moderate parameters.
  • The unconditional result implies that the set of characters producing at least one zero has density one, yet still leaves infinitely many characters free of such zeros.

Load-bearing premise

The generalized Riemann hypothesis holds for the L-functions L(s, χ χ_j) and the auxiliary L-functions arising in the proof.

What would settle it

An explicit large prime q together with fixed D_j and t such that every character χ mod q makes the product ∏ L(1/2 + it, χ χ_j) vanish would falsify the claim (assuming GRH for the relevant L-functions).

read the original abstract

In this paper, we prove the simultaneous non-vanishing of four Dirichlet $L$-functions at any point on the critical line. More precisely, let $\chi_1,\ldots,\chi_4$ be even Dirichlet characters modulo $D_1,\ldots, D_4$ respectively, where the $D_j$ are pairwise co-prime and square-free integers. Under the Generalized Riemann Hypothesis, we prove that $\prod_{j=1}^4 L(1/2+it,\chi \chi_j) \neq 0$ for a positive proportion of Dirichlet characters $\chi \pmod q$, with $q$ prime and sufficiently large in terms of the $D_j$ and $t$ (and with an explicit relationship between $D_j, t$ and $q$). Unconditionally, we also prove a simultaneous non-vanishing result for four Dirichlet $L$-functions for infinitely many characters $\chi \pmod q$, though in this case the proportion tends to zero as $q \to \infty$.

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

0 major / 3 minor

Summary. The manuscript proves simultaneous non-vanishing for the product of four Dirichlet L-functions at a fixed point 1/2 + it on the critical line. Under GRH, for q a sufficiently large prime (with explicit dependence on the fixed pairwise coprime square-free D_j and on t), a positive proportion of characters χ mod q satisfy ∏_{j=1}^4 L(1/2 + it, χ χ_j) ≠ 0, where the χ_j are even characters mod D_j. Unconditionally, the product is non-zero for infinitely many such χ, although the proportion tends to zero as q → ∞.

Significance. If the claims hold, the result strengthens the literature on non-vanishing of L-functions in character families by handling four simultaneous twists at arbitrary height t. The GRH-conditional positive-proportion statement is the stronger contribution and could feed into applications involving joint value distributions or zero statistics; the unconditional infinitude result is consistent with existing techniques but weaker. The setup with pairwise coprime square-free moduli and even characters is standard and allows clean control of conductors and functional equations.

minor comments (3)
  1. The abstract states that q is 'sufficiently large in terms of the D_j and t' with an 'explicit relationship'; the main theorem (presumably in §2 or §3) should record the precise form of this dependence, including any lower bound on the positive proportion under GRH.
  2. Clarify in the introduction whether the GRH assumption applies only to the four twisted L-functions appearing in the product or to a larger set of auxiliary L-functions used in the proof.
  3. The unconditional result is stated to have proportion tending to zero; a short remark on the rate (even if not optimal) would help readers assess its strength relative to the GRH case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of our results on the simultaneous non-vanishing of four Dirichlet L-functions. We are pleased that the referee views the GRH-conditional positive-proportion result as the stronger contribution and recognizes its potential for applications. The referee recommends minor revision, but no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a positive-proportion non-vanishing result for the product of four twisted Dirichlet L-functions at a fixed point on the critical line, conditional on the external Generalized Riemann Hypothesis. The argument proceeds via standard analytic estimates on character sums and L-function approximations under GRH, with the fixed parameters D_j (pairwise coprime square-free) and t entering only as constants that determine the size of q needed for error control. No step reduces a claimed prediction or uniqueness statement to a fitted parameter or self-citation; the unconditional infinitude result is derived from weaker bounds and is logically independent of the GRH case. The derivation chain is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; GRH is invoked as a standard domain assumption in analytic number theory for L-functions. No free parameters, invented entities, or ad-hoc axioms are mentioned in the provided text.

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Reference graph

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