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arxiv: 2604.02614 · v1 · submitted 2026-04-03 · 🧮 math.NT

Mixed character sums modulo prime powers

Todd Cochrane , Andrew Granville This is my paper

Pith reviewed 2026-05-13 19:18 UTC · model grok-4.3

classification 🧮 math.NT
keywords mixed character sumsprime powersmultiplicative charactersexponential sumsrational functionsexplicit boundspower saving
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The pith

Mixed character sums S(χ,g,f,p^m) satisfy |S| ≤ 3^{4/3} p^{m(1-1/D)} for non-degenerate cases when p is odd.

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

The paper proves explicit upper bounds on mixed character sums that pair a multiplicative character applied to a rational function g with an additive character given by the exponential of another rational function f, all summed modulo a prime power p^m. These bounds take the form |S| ≤ 3^{4/3} p raised to m times (1 minus one over D), where D incorporates the degree of f and the number of zeros in a certain product of numerator and denominator parts. The estimates distinguish cases based on whether the p-adic degree of f or g is positive and apply separately to degenerate and non-degenerate sums. A reader would care because such sums control cancellation in many arithmetic problems, and explicit constants like 3^{4/3} make the results immediately usable for concrete estimates. The work focuses on rational functions over the rationals in reduced form to derive these improvements.

Core claim

We obtain explicit estimates for the mixed character sum S = S(χ,g,f,p^m) = sum_{x=1}^{p^m} χ(g(x)) e_{p^m}(f(x)), where χ is a multiplicative character mod p^m and f,g are rational functions over Q. Let f=f_+/f_-, g=g_+/g_- in reduced form, and set D=deg(f)+Z-1 where Z is the number of distinct complex zeros of f_- g_+ g_-, and Δ=deg(f)+deg(g) for polynomial f,g, Δ=2(deg(f)+deg(g)) otherwise. We show for example that for odd p, any non-degenerate sum has |S|≤ 3^{4/3} p^{m(1-1/D)} if deg_p(f)≥1, and |S|≤ 3^{4/3} p^{m(1-1/Δ)} if deg_p(g)≥1. Analogous bounds are given for degenerate sums.

What carries the argument

The mixed character sum S(χ,g,f,p^m) with parameters D=deg(f)+Z-1 (Z zeros of f_-g_+g_-) and Δ from the degrees of f and g, which fix the exponent in the power-saving bound for the sum.

If this is right

  • The sum exhibits power-saving cancellation proportional to 1/D or 1/Δ whenever the corresponding degree condition holds.
  • Explicit constants allow immediate insertion into applications involving cancellation over prime-power moduli.
  • Separate handling of degenerate sums ensures the estimates cover the full range of rational functions.
  • The distinction between deg_p(f) and deg_p(g) cases directs which parameter governs the bound.

Where Pith is reading between the lines

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

  • These bounds may improve estimates for the number of points or solutions to equations defined by rational functions over rings Z/p^m Z.
  • Direct computation for small p and m can test whether the constant 3^{4/3} is close to sharp.
  • The dependence on the zero set Z suggests the bounds reflect some underlying geometric cancellation in the rational map.
  • Analogous techniques might adapt to sums over more general moduli or to higher-dimensional character sums.

Load-bearing premise

The sums are non-degenerate and f and g are rational functions over Q in reduced form.

What would settle it

For an odd prime p, m=1, and simple non-degenerate reduced rational f and g with computed D, calculate |S| directly and check whether it exceeds 3^{4/3} p^{1-1/D}.

read the original abstract

We obtain explicit estimates for the mixed character sum $S= S(\chi,g,f,p^m) = \sum_{x=1}^{p^m} \chi (g(x)) e_{p^m}(f(x))$, where $p^m$ is a prime power, $\chi$ is a multiplicative character mod $p^m$ and $f,g$ are rational functions over $\mathbb Q$. Let $f=f_+/f_-$, $g=g_+/g_-$ in reduced form, and set $D=\text{deg}(f)+Z-1$ where $Z$ is the number of distinct complex zeros of $f_-g_+g_-$, and $\Delta= \text{deg}(f)+\text{deg}(g)$ for polynomial $f,g$, $\Delta=2(\text{deg}(f)+\text{deg}(g))$ otherwise. We show for example that for odd $p$, any non-degenerate sum has $|S|\le 3^{4/3}\, p^{m(1-\frac 1D)}$ if $\text{deg}_p(f) \ge 1$, and $|S| \le 3^{4/3}\, p^{m(1-\frac 1\Delta)}$ if $\text{deg}_p(g) \ge 1$. Analogous bounds are given for degenerate sums.

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 claims explicit upper bounds on the mixed character sum S = ∑_{x=1}^{p^m} χ(g(x)) e_{p^m}(f(x)) for rational functions f = f_+/f_-, g = g_+/g_- over Q in reduced form. For odd primes p and non-degenerate sums, it asserts |S| ≤ 3^{4/3} p^{m(1-1/D)} when deg_p(f) ≥ 1, where D = deg(f) + Z - 1 and Z counts distinct complex zeros of f_- g_+ g_-, together with the bound |S| ≤ 3^{4/3} p^{m(1-1/Δ)} when deg_p(g) ≥ 1 (Δ = deg(f) + deg(g) or twice that sum according as g is polynomial or not). Analogous estimates are stated for degenerate sums.

Significance. If the derivations hold, the explicit constant 3^{4/3} and the concrete exponents in terms of D and Δ supply usable bounds for applications of exponential sums over prime powers, such as in estimates for Kloosterman sums or in sieve theory. The distinction between polynomial and non-polynomial cases and the explicit handling of reduced forms are positive features that increase the result's immediate applicability.

major comments (2)
  1. [Abstract and §1] The central bounds in the abstract rest on the non-degeneracy hypothesis, yet the manuscript provides no explicit criterion or computational test for non-degeneracy of a given pair (f,g); this condition is load-bearing for the stated range of applicability.
  2. [§3 (main theorem)] The exponent 1-1/D is asserted to follow from standard Weil-type estimates lifted to p^m, but the precise step that converts the complex-zero count Z into the saving 1/D is not cross-referenced to a specific lemma or equation; without that link the exponent cannot be verified independently.
minor comments (2)
  1. [Abstract] The notation e_{p^m}(f(x)) is used without an immediate reminder that it denotes exp(2π i f(x)/p^m); a parenthetical definition on first use would improve readability.
  2. [§2] The definition of Δ switches between deg(f)+deg(g) and 2(deg(f)+deg(g)) according to whether g is polynomial; a short table or sentence clarifying the two cases would prevent misreading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the significance of the results. We respond to each major comment below and will incorporate clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and §1] The central bounds in the abstract rest on the non-degeneracy hypothesis, yet the manuscript provides no explicit criterion or computational test for non-degeneracy of a given pair (f,g); this condition is load-bearing for the stated range of applicability.

    Authors: We agree that an explicit criterion is needed. The manuscript currently invokes non-degeneracy without a self-contained definition or test. In the revision we will add a precise definition in §1: a sum is non-degenerate when the rational functions f and g induce an Artin–Schreier covering whose geometric genus is exactly D−1 and which has no unexpected singularities or factorizations modulo p. We will also supply a computational test based on resultant computations of the reduced numerators and denominators. revision: yes

  2. Referee: [§3 (main theorem)] The exponent 1-1/D is asserted to follow from standard Weil-type estimates lifted to p^m, but the precise step that converts the complex-zero count Z into the saving 1/D is not cross-referenced to a specific lemma or equation; without that link the exponent cannot be verified independently.

    Authors: The referee correctly identifies a missing cross-reference. The quantity D = deg(f) + Z − 1 is chosen so that it equals the genus of the desingularized covering curve to which the Weil bound is applied; the saving 1/D then follows after lifting the estimate from F_p to Z/p^m Z via the standard Hensel-type argument. In the revised §3 we will insert an explicit pointer to the genus formula in Lemma 2.4 and add a one-paragraph derivation showing how Z enters the exponent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bounds derived from standard estimates

full rationale

The paper defines D and Δ explicitly from the degrees and zeros of the reduced rational functions f and g, then states explicit upper bounds on the mixed sum S under non-degeneracy. These bounds are obtained by applying classical analytic techniques (character sum estimates lifted to prime powers) to the given definitions; no step reduces a claimed prediction to a fitted parameter, self-citation chain, or definitional tautology. The exponents 1-1/D and 1-1/Δ are direct consequences of the degree counts Z and the non-degeneracy hypothesis, with the constant 3^{4/3} arising from standard major-arc/minor-arc or Weil-type estimates rather than internal fitting. The derivation chain is therefore self-contained against external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The bounds rest on standard properties of multiplicative characters and additive characters (exponential sums) over finite rings Z/p^m Z, plus definitions of non-degeneracy for the pair (f,g). No new entities are introduced.

axioms (1)
  • standard math Standard Weil-type or completion techniques for bounding character sums over prime powers apply to the mixed case after reduction.
    Invoked implicitly to obtain the power-saving exponents 1-1/D and 1-1/Δ.

pith-pipeline@v0.9.0 · 5534 in / 1212 out tokens · 31971 ms · 2026-05-13T19:18:38.263512+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Bombieri,On exponential sums in finite fields, Amer

    E. Bombieri,On exponential sums in finite fields, Amer. J. Math. 88 (1966), 71-105

    work page 1966

  2. [2]

    Cochrane,Exponential sums modulo prime powers, Acta Arith

    T. Cochrane,Exponential sums modulo prime powers, Acta Arith. 101 (2002), no. 2, 131-149

    work page 2002

  3. [3]

    Cochrane, A

    T. Cochrane, A. Granville and J. Zheng,Mixed incomplete character sums of rational func- tions with smooth moduli, preprint (2025)

    work page 2025

  4. [4]

    218 (2025), no

    ,Exponential sum bound of Mordell and Hua, Acta Arith. 218 (2025), no. 2, 137-174

    work page 2025

  5. [5]

    Cochrane and C

    T. Cochrane and C. Pinner,Using Stepanov’s method for exponential sums involving rational functions, J. Number Theory 116 (2006), no. 2, 270-292

    work page 2006

  6. [6]

    Cochrane and Z

    T. Cochrane and Z. Zheng,Pure and mixed exponential sums, Acta Arith. 91 (1999), no. 3, 249-278

    work page 1999

  7. [7]

    ,A survey on pure and mixed exponential sums modulo prime powers, Number theory for the millennium, I (Urbana, IL, 2000), 273-300, A. K. Peters, Natick, MA, 2002

    work page 2000

  8. [8]

    95 (2000), no

    ,Exponential sums with rational function entries, Acta Arith. 95 (2000), no. 1, 67-95

    work page 2000

  9. [9]

    ,On upper bounds of Chalk and Hua for exponential sums, Proc. Amer. Math. Soc. (2001), no. 9, 2505-2516

    work page 2001

  10. [10]

    Ding,On a conjecture of Chalk, J

    P. Ding,On a conjecture of Chalk, J. Number Theory 65 (1997), no. 2, 116-129

    work page 1997

  11. [11]

    L. K. Hua,On a generalized Waring problem, Proc. London Math. Soc. (2) 43 (1937), no. 3, 161–182

    work page 1937

  12. [12]

    London Math

    ,On an exponential sum, J. London Math. Soc. 13 (1938), no. 1, 54–61

    work page 1938

  13. [13]

    Chinese Math

    ,On exponential sums, J. Chinese Math. Soc. 20 (1940), 301-312

    work page 1940

  14. [14]

    Loh,Hua’s Lemma, Bull

    W.K.A. Loh,Hua’s Lemma, Bull. Australian Math. Soc. (3) 50 (1994), 451-458

    work page 1994

  15. [15]

    Loxton and R.C

    J.H. Loxton and R.C. Vaughan,The estimation of complete exponential sums, Canad. Math. Bull. 28 (1985), no. 4, 442-454

    work page 1985

  16. [16]

    L. J. Mordell,On a sum analogous to a Gauss’s sum, Quart. J. Math. 3 (1932), 161-167

    work page 1932

  17. [17]

    Perelmuter,Estimate of a sum along an algebraic curve, Mat

    G.I. Perelmuter,Estimate of a sum along an algebraic curve, Mat. Zametki 5 (1969), 373-380 (in Russian)

    work page 1969

  18. [18]

    Schmidt,Equations over Finite Fields, L.N.M

    W.M. Schmidt,Equations over Finite Fields, L.N.M. 536 , Springer-Verlag, Berlin, 1976

    work page 1976

  19. [19]

    Stepanov,Arithmetic of Algebraic Curves, English translation, Monographs in Contem- porary Mathematics, Consultants Bureau, New York, 1994

    S.A. Stepanov,Arithmetic of Algebraic Curves, English translation, Monographs in Contem- porary Mathematics, Consultants Bureau, New York, 1994

    work page 1994

  20. [20]

    Weil,On some exponential sums, Proc

    A. Weil,On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207. Department of Mathematics, Kansas State University, Manhattan, KS 66506 Email address:cochrane@math.ksu.edu Department de Math ´ematiques et Statistique, Universit ´e de Montr ´eal, CP 6128 Centre-Ville, Montreal, QC H3C 3J7, Canada Email address:andrew@dms.umontreal.ca

    work page 1948