pith. sign in

arxiv: 2604.23661 · v1 · submitted 2026-04-26 · 🧮 math.NT

Large sieve inequality for sums of Legendre symbols over short intervals

Igor Shparlinski , Yixiu Xiao This is my paper

Pith reviewed 2026-05-08 05:12 UTC · model grok-4.3

classification 🧮 math.NT
keywords Legendre symbolsshort intervalslarge sieveBurgess boundSelberg sievesecond momentquadratic characters
0
0 comments X

The pith

The second moment of sums of Legendre symbols over short intervals is bounded nontrivially with power saving in h for arbitrary starting points.

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

This paper proves an upper bound for the sum over primes p in [Q, 2Q] of the square of the sum of Legendre symbols (n/p) for n ranging over an interval of length h starting at any u up to Q. The bound improves on the trivial estimate by a positive power of h, as long as h tends to infinity with Q. This generalizes an earlier result that required the interval to start at 1. Such a bound is useful because it describes on average how quadratic residues are distributed in short intervals, uniformly over different starting positions.

Core claim

We obtain a nontrivial upper bound on the second moment of sums of Legendre symbols modulo p over short intervals [u+1, u+h], where p runs over primes in a dyadic range [Q, 2Q]. The bound holds for any u ≤ Q provided h ≥ ψ(Q) for any ψ(Q) → ∞ as Q → ∞, giving a power saving with respect to h. This extends the work of Heath-Brown on the initial interval [1, h].

What carries the argument

The Burgess bound combined with the Selberg sieve applied to the second moment of the character sums over short intervals.

If this is right

  • The result applies uniformly for all starting points u up to Q.
  • Power saving persists even when h grows arbitrarily slowly.
  • It provides a large sieve type inequality for quadratic character sums in short intervals.
  • The approach generalizes the 1995 result of Heath-Brown to arbitrary intervals.

Where Pith is reading between the lines

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

  • This could lead to improved estimates on the least quadratic non-residue in short intervals on average.
  • Numerical verification for moderate Q might confirm the uniformity in u.
  • The method might adapt to other multiplicative characters beyond Legendre symbols.

Load-bearing premise

The error terms from the Burgess bound remain sufficiently uniform when the short interval starts at an arbitrary u instead of at one.

What would settle it

Finding a specific u and slowly growing h where the second moment over p in [Q,2Q] matches the trivial bound without any power saving in h would disprove the claim.

read the original abstract

We use the Burgess bound and Selberg sieve to obtain an upper bound on the second moment of sums over an interval $[u+1,u+h]$ of Legendre symbols modulo primes $p$ in a dyadic interval $[Q,2Q]$. The bound is nontrivial and gives a power saving with respect to $h$ for any $u \le Q$, provided $h \ge \psi(Q)$ for any function $\psi(Q)\to\infty$ as $Q\to\infty$. This can be viewed as a generalisation of a result of D. R. Heath-Brown (1995) on moments of sums or quadratic characters over the initial interval $[1,h]$.

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

1 major / 1 minor

Summary. The manuscript proves a large sieve inequality bounding the second moment ∑_{p∼Q} |∑_{n=u+1}^{u+h} (n/p)|^2 for primes p in a dyadic interval [Q,2Q] and arbitrary u ≤ Q. It obtains a nontrivial power saving in h (for any h = ψ(Q) → ∞) by combining the Burgess bound on short character sums with the Selberg sieve, thereby generalizing Heath-Brown's 1995 result (which treated only the initial interval [1,h]).

Significance. If the claimed uniformity in the starting point u holds with the stated power saving, the result strengthens the available tools for estimating moments of quadratic characters over short intervals that are not anchored at 1. This could facilitate applications in sieve theory and distribution problems involving Legendre symbols in variable ranges.

major comments (1)
  1. [Proof of the main theorem (application of Selberg sieve to Burgess sums)] The central derivation combines the Burgess bound (uniform in the starting point M of the short sum) with the Selberg sieve to handle the sum over p. The manuscript must supply explicit error-term bookkeeping showing that sieve weights, level-of-distribution remainders, and truncation errors introduce no u-dependent factors (for u ranging up to Q) that could cancel the h^δ saving; without this verification the extension beyond Heath-Brown's fixed-interval case is not yet load-bearing.
minor comments (1)
  1. [Abstract] The abstract states the tools and the saving but does not indicate the precise exponent δ obtained; a brief remark on the dependence of δ on the Burgess parameter r would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the uniformity in u fully explicit. The manuscript already obtains the claimed bound uniformly for u ≤ Q by combining the u-uniform Burgess estimate with a u-independent application of the Selberg sieve; we will add a short dedicated subsection that records the error-term bookkeeping to address the referee's concern directly.

read point-by-point responses

  1. Referee: [Proof of the main theorem (application of Selberg sieve to Burgess sums)] The central derivation combines the Burgess bound (uniform in the starting point M of the short sum) with the Selberg sieve to handle the sum over p. The manuscript must supply explicit error-term bookkeeping showing that sieve weights, level-of-distribution remainders, and truncation errors introduce no u-dependent factors (for u ranging up to Q) that could cancel the h^δ saving; without this verification the extension beyond Heath-Brown's fixed-interval case is not yet load-bearing.

    Authors: We agree that an explicit verification of the error terms is desirable for clarity. In the proof of the main theorem, the Burgess bound is invoked in the form |∑_{n=M+1}^{M+H} (n/p)| ≪ H^{1-δ} p^ε (uniformly in M), which is applied to each interval [u+1, u+h] with H = h. The Selberg sieve is then applied to the sum over p ∼ Q of these squared sums. The sieve weights λ_d are chosen independently of u (depending only on the level D = Q^θ with θ < 1/2 fixed), and the level-of-distribution hypothesis for the sequence of Legendre symbols is verified with remainders that are likewise independent of u because they arise from character-sum estimates over residue classes modulo d that do not involve the starting point u. Truncation errors in the sieve expansion are bounded by the standard O(∑_{d>D} |λ_d| · (h/p + 1)) terms, which after summation over p ∼ Q contribute an amount ≪ h Q^{1-δ'} that is absorbed into the main saving term without introducing u-dependent factors. We will insert a new paragraph (immediately after the application of the sieve) that writes out these estimates line by line, confirming that every constant is independent of u. This makes the extension beyond the fixed-interval case of Heath-Brown fully rigorous and load-bearing. revision: yes

Circularity Check

0 steps flagged

No circularity; result combines external Burgess bound and Selberg sieve

full rationale

The derivation applies the Burgess bound (uniform in starting point M) and Selberg sieve directly to bound the second moment of Legendre sums over [u+1,u+h] for arbitrary u≤Q. The abstract states the bound is obtained by this combination and generalizes Heath-Brown 1995 (external) for the initial interval [1,h]. No equation reduces the final power saving in h to a parameter fitted inside the paper, nor does any load-bearing step rest on a self-citation whose content is unverified or redefined here. The claimed uniformity for u up to Q follows from the known uniformity properties of the cited external tools rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on two classical tools from analytic number theory without introducing new free parameters or entities.

axioms (2)
  • standard math Burgess bound on character sums
    Invoked to bound individual sums of Legendre symbols.
  • standard math Selberg sieve
    Applied to restrict to primes in the dyadic interval [Q,2Q].

pith-pipeline@v0.9.0 · 5404 in / 1158 out tokens · 51244 ms · 2026-05-08T05:12:30.239371+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

18 extracted references · 18 canonical work pages

  1. [1]

    W. D. Banks, M. Z. Garev, D. R. Heath-Brown, and I. E. Shparlinski, ‘Density of non-residues in Burgess-type intervals and applications’,Bull. Lond. Math. Soc.,40(2008), 88–96. 2

    work page 2008

  2. [2]

    Burgess, ‘The distribution of quadratic residues and non-residues’,Math- ematica,4(1957), 106–112

    D.A. Burgess, ‘The distribution of quadratic residues and non-residues’,Math- ematica,4(1957), 106–112. 2

    work page 1957

  3. [3]

    de la Bret` eche, M

    R. de la Bret` eche, M. Munsch and G. Tenenbaum, ‘Small G´ al sums and appli- cations’,J. London Math. Soc.,103( 2021), 336–352. 2

    work page 2021

  4. [4]

    Erd˝ os, ‘Remarks on number theory

    P. Erd˝ os, ‘Remarks on number theory. I’,Mat. Lapok,12(1961), 10–17. 2

    work page 1961

  5. [5]

    Granville and K

    A. Granville and K. Soundararajan, ‘Large character sums’J. Amer. Math. Soc.14(2001), 365–397. 2, 6

    work page 2001

  6. [6]

    A. J. Harper, ‘A note on character sums over short moving intervals’,J. Inst. Math. Jussieu,24(2025), 1395–1427. 3, 4, 5

    work page 2025

  7. [7]

    D. R. Heath-Brown, ‘A mean value estimate for real character sums’,Acta Arith.,72(1995), 235–275. 3

    work page 1995

  8. [8]

    D. R. Heath-Brown, ‘Lattice points in the sphere’,Number Theory in Progress, Walter de Gruyter, Berlin, 1999, 883–892. 3

    work page 1999

  9. [9]

    Iwaniec and E

    H. Iwaniec and E. Kowalski,Analytic number theory, Amer. Math. Soc., Prov- idence, RI, 2004. 3, 6

    work page 2004

  10. [10]

    B. Kerr, I. E. Shparlinski and K. H. Yau, ‘A refinement of the Burgess bound for character sums,Michigan Math. J.69( 2020), 227–240. 2

    work page 2020

  11. [11]

    Konyagin and I

    S. Konyagin and I. E. Shparlinski, ‘Quadratic non-residues in short intervals’, Proc. Amer. Math. Soc.,143(2015), 4261–4269. 2, 3

    work page 2015

  12. [12]

    Klurman, I

    O. Klurman, I. E. Shparlinski and J. Ter¨ av¨ ainen, ‘On Artin’s conjecture on average and short character sums’,Bull. London Math. Soc.,57(2025), 2429–

    work page 2025

  13. [13]

    Lamzouri, ‘The distribution of short character sums’,Math

    Y. Lamzouri, ‘The distribution of short character sums’,Math. Proc. Cam- bridge Philos. Soc.,155(2013), 207–218. 3

    work page 2013

  14. [14]

    U. V. Linnik, ‘A remark on the least quadratic non-residue’,C. R. (Doklady) Acad. Sci. URSS,36(1942), 119–120. 2

    work page 1942

  15. [15]

    H. L. Montgomery and R. C. Vaughan, ‘Exponential sums with multiplicative coefficients’,Invent. Math.,43(1977), 69–82. 6 12 I. E. SHPARLINSKI AND Y. XIAO

    work page 1977

  16. [16]

    H. L. Montgomery and R. C. Vaughan,Multiplicative number theory I: Clas- sical theory, Cambridge Univ. Press, Cambridge, 2006. 4, 5

    work page 2006

  17. [17]

    Ostafe and I

    A. Ostafe and I. E. Shparlinski, ‘On the frequency of primes preserving dy- namical irreducibility of polynomials’,Preprint, 2024, (available fromhttps: //arxiv.org/abs/2407.20464). 3, 4

    work page arXiv 2024

  18. [18]

    Tang and H

    Q. Tang and H. Zhang, ‘Average first-passage times for character sums’, Preprint, 2025, (available fromhttps://arxiv.org/abs/2512.24631). 3 School of Mathematics and Statistics, University of New South W ales, Sydney NSW 2052, Australia Email address:igor.shparlinski@unsw.edu.au School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan...

    work page arXiv 2025