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REVIEW 1 major objections 2 minor 12 references

Under GRH, the 2k-moments of Im log L(1/2+it, χ) for fixed odd squarefree conductor satisfy (C_K k L_T)^k up to k = K L_T.

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2026-06-27 14:31 UTC pith:CYCLAEIP

load-bearing objection Hughes ports the Selberg-Tsang moment bound to Im log L(1/2+it,χ) at fixed conductor under GRH, and the adaptation holds with only routine adjustments. the 1 major comments →

arxiv 2606.10242 v1 pith:CYCLAEIP submitted 2026-06-08 math.NT math.PR

A Tsang-range high-moment bound for operatorname{Im}log L(tfrac12+it,chi) under GRH

Scott D. Hughes This is my paper
classification math.NT math.PR
keywords L-functionsDirichlet charactersgeneralized Riemann hypothesishigh momentsimaginary part of log Lmean-value estimateslarge deviations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves an upper bound on the average size of |Im log L(1/2 + it, χ)| raised to the 2k power. The bound is conditional on the generalized Riemann hypothesis and holds when the conductor q is a fixed squarefree odd integer at least 3. It covers moments with k ranging up to a constant multiple of log log(qT). The result gives control on how large these imaginary parts can typically become and produces corresponding tail decay estimates.

Core claim

Conditional on GRH for L(s, χ), for every K > 0 there exist C_K and T_0 such that (1/T) ∫_T^{2T} |X_χ(t)|^{2k} dt ≤ (C_K k L_T)^k for all T ≥ T_0 and 1 ≤ k ≤ K L_T, where L_T = log log(qT), q is fixed squarefree odd ≥ 3, χ is primitive non-principal, and X_χ(t) = Im log L(1/2 + it, χ). The proof adapts Selberg's pointwise formula by splitting into three prime-power Dirichlet polynomials and applies Soundararajan's mean-value lemma to their moments.

What carries the argument

Selberg's pointwise approximate formula for the argument, adapted under GRH to X_χ(t) at fixed conductor and split into three prime-power Dirichlet polynomials whose moments are bounded via Soundararajan's mean-value lemma.

Load-bearing premise

Selberg's pointwise approximate formula for S(t) transfers directly to X_χ(t) at fixed conductor under GRH without extra error terms that would invalidate the moment calculation.

What would settle it

Numerical evaluation of the integral average of |X_χ(t)|^{2k} for T around exp(exp(10)) showing that the average exceeds any fixed multiple of (C k L_T)^k for k near K L_T would disprove the stated bound (assuming the GRH instances used in the computation hold).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Markov's inequality produces the tail bound Prob(|X_χ(t)| > V sqrt(L_T)) ≪ exp(-c V^2 / L_T) for sqrt(L_T) ≪ V ≪ L_T.
  • The same argument yields an imaginary-part version of known large-deviation upper bounds for log |ζ(1/2 + it)|.
  • The moment bound is uniform in the stated range 1 ≤ k ≤ K L_T once T exceeds T_0.

Where Pith is reading between the lines

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

  • The same splitting technique could be tested on L-functions attached to other fixed conductors or on real parts of the logarithm.
  • If the constants C_K remain bounded as K grows, the distribution of X_χ(t) would be consistent with a Gaussian of variance proportional to L_T.
  • The method supplies an explicit route to check the moment growth numerically on ranges where GRH can be verified directly.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. Conditional on GRH for L(s,χ) with fixed squarefree odd conductor q≥3 and primitive non-principal χ, the paper establishes that (1/T)∫_T^{2T} |X_χ(t)|^{2k} dt ≤ (C_K k L_T)^k for all T≥T_0 and 1≤k≤K L_T, where L_T=log log(qT) and X_χ(t)=Im log L(1/2+it,χ). The argument ports Selberg's GRH-conditional pointwise formula for S(t) to the L-function setting, splits the approximant into three prime-power Dirichlet polynomials, and evaluates the resulting moments via Soundararajan's mean-value lemma. A corollary via Markov's inequality gives the Gaussian tail bound exp(-c V^2/L_T) for √L_T ≪ V ≪ L_T.

Significance. If the central claim holds, the result supplies the first Tsang-range (k up to K log log(qT)) high-moment bound for the imaginary part of log L(1/2+it,χ) at fixed conductor, together with matching large-deviation upper bounds. This extends the corresponding zeta-function statements of Selberg-Tsang and Soundararajan to the Dirichlet L-function setting under GRH and supplies a concrete conditional model for the distribution of log L-values.

major comments (1)
  1. [The section containing the GRH-conditional approximation and three-way splitting (referenced in the abstract as the port] The L^{2k} integrability of the remainder after the three-prime-power splitting of the ported Selberg approximant to X_χ(t) is not shown to be o((k L_T)^k) uniformly for k≤K L_T. Because χ is non-principal, the remainder may retain extra oscillatory factors; without an explicit bound on its contribution to the 2k-moment integral, Soundararajan's mean-value lemma cannot be applied directly to the main terms while preserving the claimed upper bound.
minor comments (2)
  1. The dependence of C_K and T_0 on K should be stated more explicitly, including whether C_K grows with K.
  2. Notation for the three Dirichlet polynomials in the splitting should be introduced with explicit formulas rather than by reference to the zeta case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this point concerning the remainder term. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The L^{2k} integrability of the remainder after the three-prime-power splitting of the ported Selberg approximant to X_χ(t) is not shown to be o((k L_T)^k) uniformly for k≤K L_T. Because χ is non-principal, the remainder may retain extra oscillatory factors; without an explicit bound on its contribution to the 2k-moment integral, Soundararajan's mean-value lemma cannot be applied directly to the main terms while preserving the claimed upper bound.

    Authors: We agree that an explicit bound on the contribution of the remainder R(t) to the 2k-moment is required for the argument to be complete. The current manuscript sketches the three-way splitting of the GRH-conditional approximant but does not supply a uniform estimate showing (1/T)∫_T^{2T} |R(t)|^{2k} dt = o((k L_T)^k) for k ≤ K L_T. The non-principal character introduces additional phase factors that must be controlled. In the revised version we will insert a new subsection deriving such a bound directly from the GRH error term and standard mean-value estimates for the resulting short Dirichlet polynomials; the oscillations from χ will be used to obtain cancellation, but the argument will be written out in full so that Soundararajan's lemma can be applied to the three principal terms with the claimed error margin. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation ports independent external results

full rationale

The paper's proof chain consists of porting Selberg's pointwise approximate formula for S(t) to the character case under GRH, splitting into three prime-power Dirichlet polynomials, and applying Soundararajan's mean-value lemma to obtain the moment bound. These are cited external results whose statements do not depend on the target inequality for |X_χ(t)|^{2k}. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain is present in the provided abstract or reader's summary. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on GRH for the specific L-function, the portability of Selberg's approximate formula, and the applicability of Soundararajan's mean-value lemma; no free parameters or invented entities are introduced in the abstract.

axioms (3)
  • domain assumption Generalized Riemann Hypothesis for L(s,χ) with fixed conductor q
    Invoked to obtain the approximate formula for X_χ(t) and to control zero locations.
  • domain assumption Selberg's pointwise approximate formula for S(t) extends to X_χ(t) at fixed conductor
    Stated as the starting point of the proof in the abstract.
  • domain assumption Soundararajan's mean-value lemma applies to the resulting Dirichlet polynomials
    Used to evaluate the moments after splitting.

pith-pipeline@v0.9.1-grok · 5794 in / 1615 out tokens · 19384 ms · 2026-06-27T14:31:35.528067+00:00 · methodology

0 comments
read the original abstract

Conditional on the Generalized Riemann Hypothesis for $L(s,\chi)$, we prove the Selberg--Tsang high-moment bound for $X_\chi(t) = \operatorname{Im}\log L(\tfrac12+it,\chi)$ at fixed squarefree odd conductor $q \ge 3$ and primitive non-principal character $\chi$. Writing $L_T = \log\log(qT)$: for every $K > 0$ there exist constants $C_K$ and $T_0$ such that $\frac{1}{T}\int_T^{2T} |X_\chi(t)|^{2k}\,dt \le (C_K\,k\,L_T)^k$ for all $T \ge T_0$ and every integer $1 \le k \le K L_T$. The proof ports Selberg's pointwise approximate formula for $S(t)$ to $L(s,\chi)$ at fixed conductor under GRH, splits it into three prime-power Dirichlet polynomials, and evaluates their moments via Soundararajan's mean-value lemma. As a corollary, Markov's inequality yields a Gaussian-scale tail $\exp(-c V^2 / L_T)$ for $\sqrt{L_T} \ll V \ll L_T$ -- a GRH-conditional, fixed-conductor, imaginary-part analogue of the large-deviation upper bounds known for $\log|\zeta(\tfrac12+it)|$.

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 4 canonical work pages · 2 internal anchors

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