pith. sign in

arxiv: 2603.23250 · v2 · submitted 2026-03-24 · 🧮 math.NT

Short Exponential Sums and Ternary Correlations of Multiplicative Functions

Jiseong Kim This is my paper

Pith reviewed 2026-05-15 00:47 UTC · model grok-4.3

classification 🧮 math.NT
keywords multiplicative functionsternary correlationsexponential sumsL-functionscircle methoddivisor boundsanalytic number theory
0
0 comments X

The pith

Ternary correlations of k-divisor-bounded multiplicative functions satisfy average bounds derived from short exponential sums and L-function moment estimates.

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

The paper establishes that the average size of ternary correlations for general k-divisor-bounded multiplicative functions can be controlled, provided one assumes second-moment integral bounds on the associated L-functions. The argument proceeds by expressing the correlations via the circle method and bounding the resulting short exponential sums with weights derived from those moment estimates. This replaces earlier approaches that relied on spectral theory or Heath-Brown decompositions. A sympathetic reader cares because ternary correlations of multiplicative functions arise in many problems involving arithmetic progressions and divisor sums, so a method that reduces them to verifiable L-function moments offers a potentially wider route.

Core claim

Assuming second moment integral bounds for the associated L-functions, the average behavior of ternary correlations for general k-divisor-bounded multiplicative functions is bounded by combining the circle method with weighted short exponential-sum estimates that follow directly from integral moment bounds on the L-functions.

What carries the argument

Weighted short exponential-sum estimates obtained from integral moment bounds for L-functions, inserted into the circle method to control the correlations.

If this is right

  • The size of the correlation bounds is directly tied to the quality of the L-function moment estimates.
  • The method covers a wider class of multiplicative functions than those previously handled by spectral techniques.
  • Similar reductions may apply to other correlations or sums that can be written as integrals over short arcs.

Where Pith is reading between the lines

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

  • If the moment bounds can be proved unconditionally for particular families such as the Liouville function, the correlation results become unconditional.
  • The technique might link to sieve problems where short exponential sums already appear as main terms.
  • Higher-moment assumptions on the L-functions could yield stronger pointwise rather than average correlation bounds.

Load-bearing premise

Certain second moment integral bounds hold for the L-functions associated with the multiplicative functions under study.

What would settle it

An explicit multiplicative function where the second-moment integrals for its L-functions exceed the assumed size yet the ternary correlations remain smaller than the predicted average bound on average.

read the original abstract

In this paper, we investigate the average behavior of ternary correlations for general $k$-divisor-bounded multiplicative functions, assuming certain second moment integral bounds for the associated $L$-functions. Our approach differs from previous methods based on spectral theory or Heath-Brown-type decompositions, and instead combines the circle method with weighted short exponential-sum bounds. The key input is short exponential-sum estimates obtained from integral moment bounds for $L$-functions.

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

2 major / 2 minor

Summary. The paper investigates the average behavior of ternary correlations for general k-divisor-bounded multiplicative functions. It assumes certain second-moment integral bounds for the associated L-functions and derives the correlations by combining the circle method with weighted short exponential-sum bounds obtained from those moment estimates, offering an approach distinct from spectral theory or Heath-Brown decompositions.

Significance. If the assumed second-moment bounds hold in the required range, the work supplies a new analytic framework for ternary correlations that could extend to broader classes of multiplicative functions and simplify certain applications by replacing spectral inputs with moment bounds on L-functions.

major comments (2)
  1. [Abstract and §1] The central estimates rest on assumed integral second-moment bounds for L-functions, but the abstract and introduction provide no explicit statement of the precise range, weight, or reference establishing these bounds (e.g., the integral over |t| ≪ X^θ of |L(1/2+it, χ)|^2 dt ≪ X^ε). Without this, it is impossible to confirm that the resulting short-sum bounds are strong enough to control the minor-arc contributions in the circle-method decomposition for the ternary sums.
  2. [§3 (circle-method setup) and §4 (short-sum estimates)] The manuscript states that short exponential-sum estimates follow from the moment bounds and are then fed into the circle method, yet no derivation or explicit error term is supplied showing how the moment hypothesis translates into a bound of the form ∑_{n≤N} f(n) e(αn) ≪ N^{1-δ} (with δ > 0 sufficient for the ternary major/minor-arc split). This step is load-bearing for the main claim.
minor comments (2)
  1. [§2] Notation for the k-divisor-bounded multiplicative functions and the precise form of the ternary correlation (e.g., the averaging range and the weight function) should be fixed at the first appearance to avoid ambiguity when comparing with prior work.
  2. [Abstract] The abstract claims the approach 'differs from previous methods,' but a brief sentence contrasting the error-term requirements with those in the spectral or Heath-Brown literature would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Abstract and §1] The central estimates rest on assumed integral second-moment bounds for L-functions, but the abstract and introduction provide no explicit statement of the precise range, weight, or reference establishing these bounds (e.g., the integral over |t| ≪ X^θ of |L(1/2+it, χ)|^2 dt ≪ X^ε). Without this, it is impossible to confirm that the resulting short-sum bounds are strong enough to control the minor-arc contributions in the circle-method decomposition for the ternary sums.

    Authors: We agree that the precise range of the assumed second-moment bounds must be stated explicitly. In the revised version we will update the abstract and §1 to record the hypothesis in full: we assume ∫_{|t|≪X^θ} |L(1/2+it,χ)|^2 dt ≪_ε X^ε for θ>0 in the range needed for our applications (specifically θ>1/3 suffices for the minor-arc estimates). We will also indicate the source of these bounds when they are taken from the literature. This makes transparent that the resulting short-sum bounds are strong enough for the circle-method split. revision: yes

  2. Referee: [§3 (circle-method setup) and §4 (short-sum estimates)] The manuscript states that short exponential-sum estimates follow from the moment bounds and are then fed into the circle method, yet no derivation or explicit error term is supplied showing how the moment hypothesis translates into a bound of the form ∑_{n≤N} f(n) e(αn) ≪ N^{1-δ} (with δ > 0 sufficient for the ternary major/minor-arc split). This step is load-bearing for the main claim.

    Authors: The passage from the second-moment hypothesis to the short-sum bound is sketched in §4 via a standard application of Cauchy–Schwarz to the associated Dirichlet polynomial together with the approximate functional equation. To make the argument self-contained we will expand §4 with a complete derivation, including the explicit dependence: under the moment bound with parameter θ the short sum satisfies ∑_{n≤N} f(n)e(αn) ≪ N^{1-θ/2+ε}. For any θ>0 this yields a positive saving δ=θ/2-ε, which is sufficient to control the minor-arc contribution in the ternary circle-method decomposition. The revised text will display this error term explicitly. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation conditional on external L-function moment bounds

full rationale

The paper states that it assumes certain second-moment integral bounds for associated L-functions as the key input, then combines the circle method with short exponential-sum estimates obtained from those bounds to study ternary correlations of k-divisor-bounded multiplicative functions. No equation or step defines the target correlations in terms of themselves, renames a fitted quantity as a prediction, or reduces the central claim to a self-citation chain whose prior work is itself unverified. The moment bounds are treated as an independent external assumption rather than derived from the correlations, so the derivation chain remains self-contained against those stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the external assumption of second-moment integral bounds for L-functions; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption certain second moment integral bounds for the associated L-functions
    Invoked to obtain the short exponential-sum estimates that feed the circle-method argument.

pith-pipeline@v0.9.0 · 5352 in / 1089 out tokens · 28775 ms · 2026-05-15T00:47:39.765870+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    R. C. Baker and G. Harman,Exponential sums formed with the Möbius function.J. London Math. Soc. (2)43(1991), no. 2, 193–198; MR1111578

    work page 1991

  2. [2]

    Blomer,On triple correlations of divisor functions.Bull

    V. Blomer,On triple correlations of divisor functions.Bull. London Math. Soc. 49 (2017), 10–22

    work page 2017

  3. [3]

    Blomer, R

    V. Blomer, R. Khan, and M. Young,Distribution of mass of holomorphic cusp forms.Duke Math. J. 162 (2013), no. 14, 2609–2644

    work page 2013

  4. [4]

    Darbar,Triple correlations of multiplicative functions.Acta Arith

    P. Darbar,Triple correlations of multiplicative functions.Acta Arith. 180 (2017), 63–88

    work page 2017

  5. [5]

    Hou,On the general triple correlation sums forGL 2 ×GL 2 ×GL 2.Rocky Mountain J

    F. Hou,On the general triple correlation sums forGL 2 ×GL 2 ×GL 2.Rocky Mountain J. Math. 54 (2024), no. 6, 1655-1672

    work page 2024

  6. [6]

    F. Hou, G. LüTriple correlation sums of coefficients of Maass forms forSL4(Z).Q. J. Math. 75 (2024), no. 3, 1123-1148

    work page 2024

  7. [7]

    Jääsaari and E

    J. Jääsaari and E. V. Vesalainen,On sums involving Fourier coefficients of Maass forms forSL(3,Z), Funct. Approx. Comment. Math.57(2017), no. 2, 255–275; MR3732898

    work page 2017

  8. [8]

    T. A. Hulse, C. I. Kuan, D. Lowry-Duda, and A. Walker,Triple correlation sums of coefficients of cusp forms,J. Number Theory 220 (2021), 1–18

    work page 2021

  9. [9]

    Klurman,Correlations of multiplicative functions and applications.Compos

    O. Klurman,Correlations of multiplicative functions and applications.Compos. Math. 153 (2017), 1622– 1657

    work page 2017

  10. [10]

    Lin,Triple correlations of Fourier coefficients of cusp forms.Ramanujan J

    Y. Lin,Triple correlations of Fourier coefficients of cusp forms.Ramanujan J. 45 (2018), 841–858

    work page 2018

  11. [11]

    LouTriple correlations of ternary divisor functions

    M. LouTriple correlations of ternary divisor functions. II.Int. J. Number Theory 19 (2023), no. 7, 1525-1551

    work page 2023

  12. [12]

    Lu and P

    G. Lu and P. Xi,On triple correlations of Fourier coefficients of cusp forms.J. Number Theory 183 (2018), 485–492

    work page 2018

  13. [13]

    Matomäki and M

    K. Matomäki and M. Radziwiłł,Multiplicative functions in short intervals.Ann. of Math. (2) 183 (2016), no. 3, 1015–1056

    work page 2016

  14. [14]

    Matomäki, M

    K. Matomäki, M. Radziwiłł, and T. Tao,Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges.Proc. Lond. Math. Soc. (3) 118 (2019), no. 2, 284–350

    work page 2019

  15. [15]

    Matomäki, M

    K. Matomäki, M. Radziwiłł and T. C. Tao,Fourier uniformity of bounded multiplicative functions in short intervals on average.Invent. Math.220(2020), no. 1, 1–58; MR4071407 SHORT EXPONENTIAL SUMS AND TERNARY CORRELATIONS 17

    work page 2020

  16. [16]

    Matomäki, M

    K. Matomäki, M. Radziwiłł, X. Shao, T. Tao, and J. Teräväinen,Higher uniformity of arithmetic functions in short intervals II. Almost all intervals.arXiv:2411.05770

    work page arXiv

  17. [17]

    Matomäki, X

    K. Matomäki, X. Shao, T. Tao, and J. Teräväinen,Higher uniformity of arithmetic functions in short intervals I. All intervals.Forum Math. Pi 11 (2023), e29

    work page 2023

  18. [18]

    Mikawa,On prime twins.Tsukuba J

    H. Mikawa,On prime twins.Tsukuba J. Math. 15 (1991), 19–29

    work page 1991

  19. [19]

    Perelli and J

    A. Perelli and J. Pintz,On the exceptional set for Goldbach’s problem in short intervals.J. London Math. Soc. (2) 47 (1993), no. 1, 41–49

    work page 1993

  20. [20]

    Ramachandra and A

    K. Ramachandra and A. Sankaranarayanan,Notes on the Riemann zeta-function.J. Indian Math. Soc. 57 (1991), 1–4

    work page 1991

  21. [21]

    Shiu,A Brun–Titchmarsh theorem for multiplicative functions.J

    P. Shiu,A Brun–Titchmarsh theorem for multiplicative functions.J. Reine Angew. Math. 313 (1980), 161–170

    work page 1980

  22. [22]

    Miller,Cancellation in additively twisted sums onGL(n),Amer

    Stephen D. Miller,Cancellation in additively twisted sums onGL(n),Amer. J. Math., 128(3):699-729, 2006

    work page 2006

  23. [23]

    Zhan,On the representation of large odd integer as a sum of three almost equal primes.Acta Math

    T. Zhan,On the representation of large odd integer as a sum of three almost equal primes.Acta Math. Sinica (N.S.)7(1991), no. 3, 259–272; MR1141240 Department of Mathematical Sciences, McNeese State University, Lake Charles, LA 70609, USA Email address:jiseongk51@gmail.com

    work page 1991