REVIEW 4 major objections 1 minor 1 cited by
Large sieves for $\mathrm{GL}_n$ and applications
T0 review · 4 major / 1 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read This paper establishes large sieve inequalities for families of automorphic L-functions, including their inverse and logarithmic coefficients, without requiring any unproved progress toward the generalized Ramanujan conjecture, and gives th
desk verdict If the proof matches the abstract, this is a real advance in GL_n large sieves, but the ramified local coefficients for log L and L^{-1} are where I'd check first. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is a large sieve inequality: a bound on a sum over a family of L-functions of weighted Dirichlet coefficients, expressed in terms of the conductors of the family and the length of the coefficient sum. The new machinery obtains this inequality simultaneously for the coefficients of L(s,π), L(s,π)^{-1}, and log L(s,π), using the functional equation and Euler product structure of both L(s,π) and L(s,π×π0) in the conductor aspect. The key structural step is showing that the inverse and logarithmic coefficients can be controlled without the generalized Ramanujan conjecture, through uniformity in the conductor that avoids pointwise bounds on Hecke eigenvalues.
What would settle it
For a specific finite family S with known conductor distribution, evaluate both sides of the paper's large sieve inequality numerically; a single family for which the claimed inequality fails, or for which the short-sum bound is not below trivial, would refute the paper's main claim.
Extended reading notes
Core claim
The paper's central discovery is a large sieve inequality for an arbitrary finite family S of unitary cuspidal automorphic representations of GL_n over a number field. For the L-functions L(s,π) and the Rankin-Selberg products L(s,π×π0), the sieve bounds weighted sums of the Dirichlet coefficients of L, of L^{-1}, and of log L uniformly in the conductors, without invoking any unproved bound toward the generalized Ramanujan conjecture. It is the first large sieve of this type to improve on the trivial bound for short sums. The resulting inequalities give, for arbitrary S, the strongest known unconditional bound on Σ_{π∈S}|L(1/2,π)|^2, stronger zero-density estimates near Re(s)=1 for families
Load-bearing premise
The argument assumes the standard analytic properties of the automorphic and Rankin-Selberg L-functions—functional equation and Euler product—hold uniformly in the conductor aspect, especially at ramified primes; a failure of that uniformity would break the sieve's control of the implied constants.
Editorial extensions
If this is right
- Unconditional moment bounds: for any finite family S, the inequality gives the strongest known bound on Σ_{π∈S}|L(1/2,π)|^2, with no Ramanujan-type hypothesis.
- Sharper zero-density estimates: the number of possible violations of the generalized Riemann hypothesis in a thin strip near Re(s)=1 is counted more tightly for automorphic and Rankin-Selberg L-functions.
- Hypothesis-free log-free zero density: the conditional log-free zero density estimate for families of Rankin-Selberg L-functions now holds without any unproven hypotheses.
- Better counting of bad Langlands parameters: the density theorem for non-archimedean Langlands parameters is improved, making violations of the generalized Ramanujan conjecture known to be rarer.
Reading between the lines
- Because the sieve covers L^{-1} and log L coefficients, it should plug directly into mollifier constructions, which would yield unconditional non-vanishing or lower-bound results for moments of GL_n L-functions; the paper does not pursue that step.
- The removal of the Ramanujan-conjecture dependence is a structural feature, suggesting the same conductor-aspect technique may transfer to other families of L-functions (for example symmetric powers or function-field analogues) whose analytic properties are known, though the paper does not claim such a transfer.
- The short-sum improvement could be tested numerically on a concrete family, such as level-one holomorphic modular forms, where the Dirichlet coefficients are essentially divisor functions; a violation of the predicted improvement there would localize the limit of the method.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper announces large sieve inequalities for the families {L(s,π): π∈S} and {L(s,π×π0): π∈S}, where S is an arbitrary finite subset of unitary cuspidal automorphic representations of GL_n over a number field. The stated results are claimed to be independent of progress toward the generalized Ramanujan conjecture and to treat simultaneously the Dirichlet coefficients of L, L^{-1}, and log L. The abstract also claims the first such result that improves on the trivial bound for short sums, with applications to second moments at the central point, zero-density estimates near Re(s)=1, unconditional log-free zero-density estimates for Rankin-Selberg families, and a density theorem for non-archimedean Langlands parameters.
Significance. If the announced results are correct, they constitute a substantial advance: they would remove a major unproved analytic input (Ramanujan-type bounds) from a general large-sieve machinery for automorphic L-functions, extend the scope to Rankin-Selberg twists, and yield new unconditional or stronger applications. The proof is not verifiable from the abstract alone; the plausibility rests on the authors' prior record and on the apparent continuity with existing analytic-conductor large sieves. The paper would be of high interest to analytic number theorists working on automorphic L-functions, moments, and zero-density estimates.
major comments (4)
- [Abstract, first paragraph] The main theorem is stated only in prose; the precise hypotheses are missing. In particular, for the large sieve inequalities to be meaningful one needs to know the analytic conductor parameter Q(S), the dependence of the implied constants on n, n0, the number field F, and the allowable range of |S| and N. Without these hypotheses and quantifications, the central claim 'independent of progress towards the generalized Ramanujan conjecture' cannot be independently checked. The full manuscript must state the exact form of the inequality, including the power of Q and the admissible ranges.
- [Abstract, first paragraph (local factors at ramified primes)] For the families L(s,π) and especially L(s,π×π0), the coefficients of L^{-1} and log L at primes ramified in π or π0 are not simple Hecke eigenvalues; they derive from local zeta integrals and may depend on the conductor exponent. A load-bearing point is whether these coefficients satisfy uniform bounds that do not use the generalized Ramanujan conjecture. If the proof bounds them by a crude p^{fθ} with f the local conductor exponent, the saving supplied by the large sieve could be absorbed by ramified places, and the claimed improvement over the trivial Ramanujan-dependent bound could fail. The manuscript should explicitly display the local coefficient bounds and show that the conductor-weighting in the sieve is preserved.
- [Abstract, short-sum improvement] The claim that this is 'the first such result that improves upon the trivial bound for short sums' is not assessable from the abstract because the term 'short sums' is not defined, nor is the comparison with the trivial bound stated. To verify this claim, the paper must specify the length N relative to the analytic conductor Q and the size of S for which the improvement holds, and give the resulting exponent beyond the trivial bound. The abstract currently leaves open the possibility that the improvement occurs only in a narrow parameter regime.
- [Abstract, applications] The applications listed are presented without quantitative statements. Since the entire evaluation of the paper's contribution depends on the strength of these consequences, the manuscript should provide precise statements for at least the key applications: the exact bound for ∑|L(1/2,π)|^2 for arbitrary S, the zero-density exponent in the GRH-violation estimate, and the unconditional log-free zero-density estimate. Without these, the significance of the results relative to Brumley-Thorner-Zaman and Lichtman-Pascadi cannot be measured.
minor comments (1)
- [Abstract, general] The abstract would be more useful if it identified theorem numbers corresponding to each announced result and stated the main analytic conductor parameter explicitly. This is a presentation issue, but it would aid the reader substantially.
Circularity Check
No circularity detectable from the abstract; claims are external and build on prior work only as benchmarks to improve.
full rationale
This is an abstract-only review. The paper claims new large sieve inequalities for automorphic L-functions and Rankin--Selberg L-functions that are independent of the generalized Ramanujan conjecture, handle L, L^{-1}, and log L, and improve on trivial bounds for short sums. The applications listed are improvements of existing results by Brumley--Thorner--Zaman and Lichtman--Pascadi, and the appearance of the present authors on those prior results is not circular: the abstract presents the new results as strengthening, not as consequences of, those prior works. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported, and no ansatz is smuggled in via self-citation. Because the abstract contains no equations or derivation steps, there is no specific reduction to exhibit, and no circularity can be substantiated under the hard rules. The plausible risks, such as uniformity of local factors at ramified primes, are matters of mathematical correctness, not circularity.
Assumptions & free parameters
assumptions (3)
- domain assumption Standard analytic properties of automorphic L-functions on GL_n: meromorphic continuation, functional equation, Euler product over finite places.
- domain assumption Rankin-Selberg L-functions L(s, π × π0) also satisfy the standard analytic properties and are automorphic.
- domain assumption Uniform bounds for the number of representations with a given conductor in the large sieve.
Cite this review
Pith. "Pith review of Large sieves for $\mathrm{GL}_n$ and applications." pith.science (2026). https://pith.science/paper/BM5SURY6
@misc{pith2026250814888,
author = {Pith},
title = {Pith review of: Large sieves for $\mathrmGL_n$ and applications},
year = {2026},
howpublished = {\url{https://pith.science/paper/BM5SURY6}},
note = {Machine review of arXiv:2508.14888}
}
abstract
Let $\mathfrak{F}_n$ be the set of unitary cuspidal automorphic representations of $\mathrm{GL}_n$ over a number field $F$, and let $S\subseteq\mathfrak{F}_n$ be an arbitrary finite subset. Given $\pi_0\in\mathfrak{F}_{n_0}$, we establish large sieve inequalities for the families $\{L(s,\pi)\colon \pi\in S\}$ and $\{L(s,\pi\times\pi_0)\colon \pi\in S\}$ that, unlike previous results, are independent of progress towards the generalized Ramanujan conjecture, and simultaneously handle the Dirichlet coefficients of $L$, $L^{-1}$, and $\log L$. We also give the first such result that improves upon the trivial bound for short sums. We present several applications, including: (1) the strongest bound for $\sum_{\pi\in S}|L(\frac{1}{2},\pi)|^2$ that holds for arbitrary $S$, (2) significant improvements to zero density estimates for families of automorphic and Rankin--Selberg $L$-functions, counting violations to the generalized Riemann hypothesis near $\mathrm{Re}(s)=1$, (3) the removal of all unproven hypotheses in the conditional log-free zero density estimate for families of Rankin--Selberg $L$-functions proved by Brumley, Thorner, and Zaman, and (4) an improvement of the density theorem for non-archimedean Langlands parameters due to Lichtman and Pascadi, counting violations to the generalized Ramanujan conjecture.
Forward citations
Cited by 1 Pith paper
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Power sums and Siegel-type zero-free regions for L-functions
For every epsilon>0 and fixed pi', L(s,pi) and L(s,pi×pi') are claimed to be lower bounded and zero-free in a c C_pi^{-epsilon} neighborhood of Re(s)=1, with ineffective c.
Reviewed August 5, 2026 · model on record in the stance chip above.
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