Unshared zeros of Dirichlet L-functions

2026/07/24 by William Banks, Kyle Loftus
#math.NT

paper · pdf

Abstract

We prove that no Dirichlet L-function (and more generally, no nontrivial finite linear combination of Dirichlet L-functions) can vanish at every zero of a fixed L(s,χ₀). At the heart of the proof is a short-window asymptotic for the twisted discrete moment ∑_ρ x^ρ L(ρ,χ₁), where χ₀≠χ₁ are primitive Dirichlet characters, ρ=β+iγ runs over zeros of L(s,χ₀) with T-Δ<γ≤ T, and x∈ℤ. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width Δ∈[T e^{-C√(log T)}, T/log T], thus reaching windows shorter than T(log T)^-A for any fixed A. Notably, the main term (χ₁(x))/(2π) Δlog T depends on x only through the single value χ₁(x). Since distinct primitive characters are distinguished by their values, varying x isolates the contribution of each L-function within a linear combination, and we deduce that for a positive density of x∈ℕ, every nontrivial combination is nonzero at some zero of L(s,χ₀) in any sufficiently high short window. The proof combines contour integration of -(L')/(L)(1-s,χ₀) L(s,χ₁) with short-interval estimates for the Dirichlet convolution (χ₀Λ)*χ₁, which derive from the classical de la Vallée Poussin zero-free region.

Citations

• The Generalized Riemann Hypothesis from zeros of a single L-function
• Towards the Generalized Riemann Hypothesis using only zeros of the Riemann zeta function
• On the Value Distribution of Two Dirichlet L-functions
• On the zeros of the Riemann Zeta function
• The distribution of prime numbers

Related

• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189
• Simple zeros of primitive Dirichlet L-functions and the asymptotic large sieve
• Zeros of Dirichlet L-functions near the critical line
• Distinct zeros and simple zeros of Dirichlet L-functions
• Critical zeros of Dirichlet L-functions