pith. sign in

arxiv: 2208.03358 · v2 · submitted 2022-08-05 · 🧮 math.NT

The large sieve for self-dual Eisenstein series of varying levels

Matthew P Young This is my paper

Pith reviewed 2026-05-24 10:57 UTC · model grok-4.3

classification 🧮 math.NT
keywords large sieveEisenstein seriesself-dualvarying levelsrationals by heightrecursive methodanalytic number theory
0
0 comments X

The pith

An essentially optimal large sieve inequality holds for self-dual Eisenstein series of varying levels.

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

The paper establishes a large sieve bound for self-dual Eisenstein series where the level varies. This bound is essentially optimal and matches the expected size from the number of such series. The result can also be viewed as a large sieve inequality for rational numbers ordered by their height. A recursive proof method is used that builds on earlier large sieve techniques without losing the optimality. Readers care because such inequalities control correlations among these objects and serve as tools for arithmetic statistics.

Core claim

We prove an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. This bound can alternatively be interpreted as a large sieve inequality for rationals ordered by height. The method of proof is recursive, and has some elements in common with Heath-Brown's quadratic large sieve, and the asymptotic large sieve of Conrey, Iwaniec, and Soundararajan.

What carries the argument

The recursive method that extends prior large sieve techniques to self-dual Eisenstein series of varying levels without incurring losses.

If this is right

  • The inequality controls the size of sums over these series up to the expected main term.
  • The same bound applies directly to rationals when they are ordered by height.
  • The recursive approach handles the variation in level without introducing extra factors that would weaken the result.
  • The bound is consistent with the dimension of the space of such series at each level.

Where Pith is reading between the lines

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

  • The recursive technique might adapt to other families of automorphic forms with varying conductors.
  • The rationals interpretation could link the inequality to problems in Diophantine approximation or geometry of numbers.
  • Numerical checks on small levels could test the implied constants in the bound.
  • The method may suggest ways to obtain large sieves for related objects like Maass forms.

Load-bearing premise

The recursive method extends prior large sieve techniques to this setting without incurring losses that would prevent the bound from being essentially optimal.

What would settle it

An explicit computation for a sequence of levels showing that the left-hand side of the inequality exceeds the right-hand side by more than a fixed constant multiple.

read the original abstract

We prove an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. This bound can alternatively be interpreted as a large sieve inequality for rationals ordered by height. The method of proof is recursive, and has some elements in common with Heath-Brown's quadratic large sieve, and the asymptotic large sieve of Conrey, Iwaniec, and Soundararajan.

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 / 2 minor

Summary. The manuscript proves an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. Equivalently, this yields a large sieve inequality for rational numbers ordered by height. The proof proceeds recursively and incorporates elements of Heath-Brown's quadratic large sieve together with the asymptotic large sieve of Conrey-Iwaniec-Soundararajan.

Significance. If the central theorem holds, the result supplies a sharp bound in the large-sieve literature for automorphic forms at varying levels, extending prior techniques without apparent loss of optimality. The recursive structure, when it succeeds in preserving the expected main term, constitutes a technical advance that may apply to related distribution problems in analytic number theory.

minor comments (2)
  1. The abstract states the result clearly, but the introduction would benefit from an early, self-contained statement of the main theorem (including the precise form of the large-sieve constant and the range of the level parameter) before the recursive argument is developed.
  2. Notation for the self-dual Eisenstein series and the height function on rationals should be fixed consistently from the first appearance; a short table of symbols would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the main result and the recommendation for minor revision. No major comments appear in the report, so there are no specific points requiring point-by-point response at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent prior techniques

full rationale

The paper's central result is an essentially optimal large sieve inequality proved via a recursive method that explicitly extends independent prior results by Heath-Brown and by Conrey-Iwaniec-Soundararajan. The abstract states the method 'has some elements in common with' those works, indicating the derivation chain incorporates external techniques rather than reducing to self-defined quantities, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or steps in the provided description exhibit self-definitional closure or renaming of known results as new derivations. The claim is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or ad-hoc axioms are mentioned. The proof relies on standard properties of Eisenstein series and large sieve methods from prior literature.

axioms (1)
  • standard math Standard analytic properties of self-dual Eisenstein series and large sieve inequalities established in prior work
    The recursive method builds directly on these background results.

pith-pipeline@v0.9.0 · 5573 in / 1195 out tokens · 24328 ms · 2026-05-24T10:57:21.811354+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 3 internal anchors

  1. [1]

    Blomer and J

    V. Blomer and J. Buttcane, On the subconvexity problem for L -functions on GL(3)

    work page

  2. [2]

    Blomer and J

    V. Blomer and J. Buttcane, Global decomposition of GL(3) Kloosterman sums and the spectral large sieve. J. Reine Angew. Math. 757 (2019), 51--88

    work page 2019

  3. [3]

    Bombieri and H

    E. Bombieri and H. Iwaniec, On the order of ( 1 2 +it) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 3, 449--472

    work page 1986

  4. [4]

    Bombieri and H

    E. Bombieri and H. Iwaniec, Some mean-value theorems for exponential sums. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 3, 473--486

    work page 1986

  5. [5]

    J. B. Conrey, H. Iwaniec, and K. Soundararajan, Asymptotic Large Sieve, arXiv:1105.1176

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Deshouillers and H

    J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms . Invent. Math. 70 (1982/83), no. 2, 219--288

    work page 1982

  7. [7]

    W. Duke, J. B. Friedlander, and H. Iwaniec, Bilinear forms with Kloosterman fractions. Invent. Math. 128 (1997), no. 1, 23--43

    work page 1997

  8. [8]

    Duke, and E

    W. Duke, and E. Kowalski,

    work page

  9. [9]

    Ellenberg, C

    J. Ellenberg, C. Elsholtz, C. Hall, and E. Kowalski, Non-simple abelian varieties in a family: geometric and analytic approaches. J. Lond. Math. Soc. (2) 80 (2009), no. 1, 135--154

    work page 2009

  10. [10]

    Forti and C

    M. Forti and C. Viola, On large sieve type estimates for the Dirichlet series operator. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 31--49. Amer. Math. Soc., Providence, R.I., 1973

    work page 1972

  11. [11]

    Gelbart and H

    S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3) . Ann. Sci. \' E cole Norm. Sup. (4) 11 (1978), no. 4, 471--542

    work page 1978

  12. [12]

    Goldfeld, Automorphic forms and L -functions for the group GL (n, R )

    D. Goldfeld, Automorphic forms and L -functions for the group GL (n, R ) . With an appendix by Kevin A. Broughan. Cambridge Studies in Advanced Mathematics, 99. Cambridge University Press, Cambridge, 2006

    work page 2006

  13. [13]

    Heath-Brown, A mean value estimate for real character sums

    D.R. Heath-Brown, A mean value estimate for real character sums. Acta Arith. 72 (1995), no. 3, 235--275

    work page 1995

  14. [14]

    Hooley, On the Barban-Davenport-Halberstam theorem

    C. Hooley, On the Barban-Davenport-Halberstam theorem. I. J. Reine Angew. Math. 274(275) (1975), 206--223

    work page 1975

  15. [15]

    Iwaniec, Topics in Classical Automorphic Forms, Grad

    H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math., vol 17, Amer. Math. Soc., 1997

    work page 1997

  16. [16]

    Iwaniec and E

    H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004

    work page 2004

  17. [17]

    Iwaniec and Xiaoqing Li, The orthogonality of Hecke eigenvalues

    H. Iwaniec and Xiaoqing Li, The orthogonality of Hecke eigenvalues. Compos. Math. 143 (2007), no. 3, 541--565

    work page 2007

  18. [18]

    Jutila and Y

    M. Jutila and Y. Motohashi,

    work page

  19. [19]

    Khan and M

    R. Khan and M. Young,

    work page

  20. [20]

    Kowalski, The large sieve and its applications

    E. Kowalski, The large sieve and its applications. Arithmetic geometry, random walks and discrete groups. Cambridge Tracts in Mathematics, 175. Cambridge University Press, Cambridge, 2008. xxii+293 pp

    work page 2008

  21. [21]

    Lapid and W

    E. Lapid and W. M\" u ller,

    work page

  22. [22]

    Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms

    W. Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms. Acta Arith. 70 (1995), no. 4, 377--391

    work page 1995

  23. [23]

    Montgomery, Topics in multiplicative number theory

    H. Montgomery, Topics in multiplicative number theory. Lecture Notes in Mathematics, Vol. 227. Springer-Verlag, Berlin-New York, 1971. ix+178 pp

    work page 1971

  24. [24]

    Thorner and A

    J. Thorner and A. Zaman, An unconditional GL_n large sieve,

    work page

  25. [25]

    Young, Bilinear forms with GL_3 Kloosterman sums and the spectral large sieve

    M. Young, Bilinear forms with GL_3 Kloosterman sums and the spectral large sieve

    work page

  26. [26]

    An improved spectral large sieve inequality for $SL_3(\mathbb{Z})$

    M. Young, An improved spectral large sieve inequality for SL_3( Z ) . Preprint, 2021, arXiv:2102.02796

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  27. [27]

    Hilbert's irreducibility theorem and the larger sieve

    D. Zywina, Hilbert's irreducibility theorem and the larger sieve. arXiv:1011.6465

    work page internal anchor Pith review Pith/arXiv arXiv