REVIEW
Unshared zeros of Dirichlet $L$-functions
Number Theory · Submitted 24 Jul 2026 · 31 references
Pith's verdict: Genuinely new short-window twisted moments, but the proof's displayed kernel drops the needed conjugate; restore the overbar and it likely goes through.
T0 · AI review
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Referee report
The paper
The paper claims
No nontrivial finite linear combination of distinct Dirichlet L-functions can vanish at every zero of a fixed Dirichlet L-function.
The reading
What carries the argument
The contour integral of x^s times K(s) = −L'/L(1−s, χ0̄) L(s, χ1) around a rectangle of height Δ at height T. The poles of K inside the rectangle are exactly the zeros ρ of L(s, χ0), with residue m_ρ x^ρ L(ρ, χ1), so Cauchy's theorem reproduces the left-hand side of the asymptotic. On the right edge, the functional equation for the logarithmic derivative converts the factor into −L'/L(s, χ0) plus explicit lower-order terms; the n = x term of the Dirichlet series produces the main term. The left edge is controlled by a short-interval bound, Lemma 2.3, for the Dirichlet convolution k = (χ0Λ) * χ1, which is where the classical zero-free region enters and sets the lower limit on Δ.
Extended reading notes
Core claim
Equation (1.1) states that for distinct primitive characters χ0 and χ1 with moduli at most Q, every integer x ≥ 10, every sufficiently large T, and every window width Δ between T exp(−C√log T) and T/log T, the sum over zeros ρ = β + iγ of L(s, χ0) with T−Δ < γ ≤ T of x^ρ L(ρ, χ1) equals (χ1(x)/(2π)) Δ log T + O(Δ). This is the first short-window version of such moments, reaching windows shorter than T(log T)^{-A} for every fixed A. From it, Corollary 1.2 follows: for a positive density of integers x, every nontrivial finite linear combination of distinct Dirichlet L-functions is nonzero at some zero of L(s, χ0) in any sufficiently high short window.
Load-bearing premise
The proof needs the short-interval sums of the arithmetic function k = (χ0Λ) * χ1 to be one logarithm smaller than the trivial bound; if Lemma 2.3's O(δ) estimate failed for some admissible pair of characters, the left-edge integral would contribute a term of the same size as the main term.
Editorial extensions
If this is right
- No single Dirichlet L-function can vanish at every zero of another distinct Dirichlet L-function.
- No nontrivial finite linear combination of distinct Dirichlet L-functions can vanish at every zero of a fixed L(s, χ0).
- The failure of shared zeros occurs in arbitrarily short windows, as short as T exp(−C√log T), not merely as a global density statement.
- For a positive density of integers x, the combination is nonzero at some zero in each sufficiently high short window.
- The proof is unconditional, using only the classical zero-free region rather than any generalized Riemann hypothesis.
Reading between the lines
- The same contour method likely extends to derivatives of Dirichlet L-functions, predicting analogous 'no shared zeros' statements for derivative values at zeros.
- Because the main term depends on x through the single value χ1(x), numerical experiments could test whether the error term's dependence on Q and x matches the implied constants in the proof.
- The mechanism may adapt to other L-functions with a functional equation of the same shape, provided the corresponding convolution analogue satisfies short-interval cancellation.
Original 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 $L(s,\chi_0)$. At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$, where $\chi_0\ne\chi_1$ are primitive Dirichlet characters, $\rho=\beta+i\gamma$ runs over zeros of $L(s,\chi_0)$ with $T-\Delta<\gamma\le T$, and $x\in\mathbb{Z}$. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\Delta\in[T e^{-C\sqrt{\log T}},\,T/\log T]$, thus reaching windows shorter than $T(\log T)^{-A}$ for any fixed $A$. Notably, the main term $\frac{\chi_1(x)}{2\pi}\,\Delta\log T$ depends on $x$ only through the single value $\chi_1(x)$. 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 $x\in\mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,\chi_0)$ in any sufficiently high short window. The proof combines contour integration of $-\frac{L'}{L}(1-s,\overline{\chi}_0)\,L(s,\chi_1)$ with short-interval estimates for the Dirichlet convolution $(\chi_0\Lambda)*\chi_1$, which derive from the classical de la Vall\'ee Poussin zero-free region.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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 $L(s,\chi_0)$. At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$, where $\chi_0\ne\chi_1$ are primitive Dirichlet characters, $\rho=\beta+i\gamma$ runs over zeros of $L(s,\chi_0)$ with $T-\Delta<\gamma\le T$, and $x\in\mathbb{Z}$. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\Delta\in[T e^{-C\sqrt{\log T}},\,T/\log T]$, thus reaching windows shorter than $T(\log T)^{-A}$ for any fixed $A$. Notably, the main term $\frac{\chi_1(x)}{2\pi}\,\Delta\log T$ depends on $x$ only through the single value $\chi_1(x)$. 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 $x\in\mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,\chi_0)$ in any sufficiently high short window. The proof combines contour integration of $-\frac{L'}{L}(1-s,\overline{\chi}_0)\,L(s,\chi_1)$ with short-interval estimates for the Dirichlet convolution $(\chi_0\Lambda)*\chi_1$, which derive from the classical de la Vall\'ee Poussin zero-free region.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $L(s,\chi_0)$. | result 0.96 | 0.96 | 0.96 | asserted 1.00 |
| s1 | At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$, where $\chi_0\ne\chi_1$ are primitive Dirichlet characters, $\rho=\beta+i\gamma$ runs over zeros of $L(s,\chi_0)$ with $T-\Delta<\gamma\le T$, and $x\in\mathbb{Z}$. | method 0.86 | 0.92 | 0.05 | asserted 0.99 |
| s2 | The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\Delta\in[T e^{-C\sqrt{\log T}},\,T/\log T]$, thus reaching windows shorter than $T(\log T)^{-A}$ for any fixed $A$. | result 0.81 | 0.90 | 0.04 | asserted 1.00 |
| s3 | Notably, the main term $\frac{\chi_1(x)}{2\pi}\,\Delta\log T$ depends on $x$ only through the single value $\chi_1(x)$. | result 0.90 | 0.89 | 0.03 | asserted 1.00 |
| s4 | 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 $x\in\mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,\chi_0)$ in any sufficiently high short window. | result 0.63 | 0.94 | 0.42 | asserted 1.00 |
| s5 | The proof combines contour integration of $-\frac{L'}{L}(1-s,\overline{\chi}_0)\,L(s,\chi_1)$ with short-interval estimates for the Dirichlet convolution $(\chi_0\Lambda)*\chi_1$, which derive from the classical de la Vall\'ee Poussin zero-free region. | method 0.99 | 0.80 | 0.02 | asserted 0.99 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2607.22930 by the authors.
References
Discussion
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