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arxiv: 2607.01184 · v1 · pith:EA5TG5MVnew · submitted 2026-07-01 · 🧮 math.NT

Lower bounds for low moments of character sums, I: Short sums with general multiplicative weights

Adam J. Harper This is my paper

Pith reviewed 2026-07-02 06:42 UTC · model grok-4.3

classification 🧮 math.NT
keywords Dirichlet characterscharacter sumsmomentslower boundsPerron integralmultiplicative weightszeta sums
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The pith

The paper establishes sharp lower bounds for the 2q-th moments of short Dirichlet character sums that exactly match the author's previous upper bounds showing better than square-root cancellation.

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

This work proves lower bounds for the average of |sum_{n≤x} χ(n)|^{2q} over characters mod r, for r prime and x up to r^{0.499}, with 0≤q≤1. These bounds are sharp and match the upper bounds from prior work that demonstrated more cancellation than the naive square-root size. The result also applies to moments of partial sums of the zeta function on the critical line and to character sums twisted by bounded multiplicative functions. A reader would care because these moments control the typical behavior of character sums, which appear throughout analytic number theory in problems like the distribution of primes. The proof relies on comparing three different averages involving a barrier-adjusted Perron integral inspired by random multiplicative functions.

Core claim

We establish sharp lower bounds for the Dirichlet character moments 1/(r-1) sum_χ |sum_{n≤x} χ(n)|^{2q}, where r is a large prime, 1≤x≤r^{0.499}, and 0≤q≤1 is real. These match the better than square-root cancellation upper bounds obtained in previous work of the author. We prove the same sharp lower bounds for the moments 1/T ∫ |sum_{n≤x} n^{it}|^{2q} dt of zeta sums, and more generally for moments of character sums sum_{n≤x} h(n) χ(n) with suitably bounded multiplicative twist h(n).

What carries the argument

A comparison of the sizes of the three averaged quantities (1/(r-1)) sum_χ (sum χ(n)) conj(I(χ)), (1/(r-1)) sum |I(χ)|^2 and (1/(r-1)) sum |I(χ)|^4, where I(χ) is the barrier-adjusted Perron integral.

If this is right

  • The moments of the short character sums are at least a positive constant times x^q on average.
  • The same sharp lower bounds hold for the moments of the zeta sums sum_{n≤x} n^{it}.
  • The same sharp lower bounds hold for the moments of the twisted sums sum_{n≤x} h(n) χ(n) with bounded multiplicative h.
  • The approach extends in a companion paper to x up to 0.99r, giving a positive proportion non-vanishing result for Dirichlet theta functions.

Where Pith is reading between the lines

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

  • The method based on the barrier-adjusted Perron integral may apply to moments in other ranges of x or to sums over other arithmetic functions.
  • The matching of upper and lower bounds suggests that the typical size of these character sums is governed by the same mechanism as for random multiplicative functions.
  • The non-vanishing result for theta functions in the companion paper is a direct consequence of the moment lower bounds established here.

Load-bearing premise

The comparison of the sizes of the three averaged quantities is valid for the stated range of x and q.

What would settle it

A calculation or computation showing that for some x and q in the range, one of the three averaged quantities is not comparable in size to the others in the asserted way.

read the original abstract

We establish sharp lower bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r^{0.499}$, and $0 \leq q \leq 1$ is real. These match the better than squareroot cancellation upper bounds obtained in previous work of the author. We prove the same sharp lower bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of zeta sums, and more generally for moments of character sums $\sum_{n \leq x} h(n) \chi(n)$ with suitably bounded multiplicative twist $h(n)$. The proofs are based on a comparison of the sizes of $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} (\sum_{n \leq x} \chi(n)) \overline{I(\chi)}$, $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |I(\chi)|^2$ and $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |I(\chi)|^4$, where $I(\chi)$ is a certain ``barrier adjusted'' Perron integral inspired by the analogous results for random multiplicative functions. In a companion paper, we extend these arguments to the full interesting range $x \leq 0.99r$ for the unweighted character sum moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$. This leads to a positive proportion non-vanishing result for Dirichlet theta functions $\theta(1,\chi)$.

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 / 3 minor

Summary. The paper proves sharp lower bounds for the moments (1/(r-1)) ∑_χ |∑_{n≤x} χ(n)|^{2q} where r is prime, x ≤ r^{0.499}, and 0 ≤ q ≤ 1 real. The bounds match the author's prior upper bounds that exhibit better than square-root cancellation. The same lower bounds are obtained for the moments of zeta sums (1/T) ∫ |∑_{n≤x} n^{it}|^{2q} dt and for twisted sums ∑_{n≤x} h(n) χ(n) with bounded multiplicative h. The argument proceeds by showing that three averaged quantities—the inner product of the character sum with a barrier-adjusted Perron integral I(χ), the L^2 norm of I(χ), and the L^4 norm of I(χ)—are of comparable size; this comparison is transferred from the setting of random multiplicative functions and is asserted to hold precisely in the stated short-sum range.

Significance. If the comparison of the three averaged quantities holds, the paper supplies matching lower bounds that complete the asymptotic picture for these low moments in the short range x ≤ r^{0.499}. The technique of comparing moments via a barrier-adjusted Perron integral, adapted from random-multiplicative-function results, is a clear methodological strength; the restriction of the range is chosen exactly so that the approximation and moment comparison remain valid. The companion paper's extension to x ≤ 0.99 r and the resulting positive-proportion non-vanishing for Dirichlet theta functions are natural consequences.

minor comments (3)
  1. The definition and normalization of the barrier-adjusted Perron integral I(χ) should be stated explicitly in the introduction (or as Eq. (1.3)) rather than deferred to the technical sections, to make the comparison of the three averaged quantities immediately legible.
  2. The precise error terms arising from the comparison of the three averaged quantities (the inner-product term, ||I||_2^2 and ||I||_4^4) are not quantified in the abstract or introduction; adding a short display of the admissible error range would clarify why the argument stops at x ≤ r^{0.499}.
  3. A brief remark on how the constants in the random-multiplicative-function model are transferred to the deterministic character-sum setting (without re-deriving them) would help readers assess the dependence on the author's earlier upper-bound work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation establishes lower bounds via direct comparison of the three averaged quantities (the cross term with I(χ), |I(χ)|^2, and |I(χ)|^4) for the stated range of x and q. This comparison is asserted on the basis of the barrier-adjusted Perron integral construction and is not shown to reduce to a fitted parameter, self-definition, or load-bearing self-citation chain. The matching to prior upper bounds by the same author is a statement of sharpness rather than part of the lower-bound derivation itself; the technique is described as inspired by random-multiplicative-function results without importing an unverified uniqueness theorem or ansatz that forces the target result. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; the central claim rests on the validity of the size comparison between the three averaged quantities built from the barrier-adjusted Perron integral, but no explicit free parameters, axioms, or invented entities are visible in the provided text.

axioms (1)
  • domain assumption The comparison of the three averaged quantities (sum χ(n) conj(I(χ)), |I(χ)|^2, |I(χ)|^4) determines the lower bound on the 2q-moment
    The proof strategy is described as relying on this comparison; details of why the comparison holds are not supplied in the abstract.

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Reference graph

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