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Counterexamples of Friedlander--Iwaniec dual sums conjecture

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Pith's one-line read The Friedlander–Iwaniec dual-sum conjecture fails for every zeta power of degree m≥4.

desk verdict A clean, elementary endpoint-resonance construction that likely refutes Friedlander–Iwaniec's uniform subpower bound for dual sums of ζ(s)^m, with the main uncertainty being the exact scope of the original conjecture. read the letter →

arxiv 2607.16695 v1 pith:PCF2GGCY submitted 2026-07-18 math.NT

Khai-Hoan Nguyen-Dang This is my paper
classification math.NT MSC 11M4111L0711N3711F66
keywords Friedlander–Iwaniecconjecturenonlineardualsumssharptruncationzetafunctionpowersdivisorendpointresonancesubpowerboundsexponential
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs explicit counterexamples to a 2005 conjecture predicting that sharply truncated nonlinear dual sums of L-function coefficients are bounded by a subpower factor uniformly in the truncation point and the external variable. Taking both Dirichlet series to be the m-th power of the Riemann zeta function with m≥4, the author tunes the external variable x so that a block of the last H terms near the cutoff N all have the same sign. Because the m-fold divisor numbers satisfy d_m(n)≥1, this block has mass N^Δ with Δ>0, contradicting the conjectured N^ε bound. The failure persists even in the printed range N≤x for m≥6.

What carries the argument

The sharp dual sum B_{ℓ,D}(x,N) and a general endpoint-sector lemma (Theorem 2.1). For a phase Φ(n)=κ(nx)^ν with 0<ν<1, the derivative at n=N is of size N^{ν(a+1)-1} when x≍N^a, so the phase changes by only a bounded amount on a block of reciprocal-derivative length H≍N^{1-ν(a+1)}. Choosing x so the terminal phase hits a non-zero point of the periodic profile forces the whole block to align; with coefficients bounded below by c*>0, the block's weighted sum is power-sized. Applied to ζ(s)^m, the coefficients are d_m(n)≥1, ν=1/m, and the profile is the cosine.

What would settle it

Consult the original statement of the conjecture to verify that a degree-m Dirichlet series with a pole of order m≥4 at s=1 and κ_j=0 is explicitly admitted; alternatively, numerically compute B_4(x_N,N) for m=4, a=0.4, λ=1 for large N and check whether its magnitude consistently exceeds, say, N^{0.02} as predicted.

Watch

Extended reading notes

Core claim

Theorem 1.2 and Corollary 1.3: for every integer m≥4 there exist x_N≍N^a with 0<a<(m-3)/2 and M_N∈{N-H_N,N} such that |B_m(x_N,M_N)| ≫ N^{(m-3-2a)/(2m)}. Since the exponent is positive, the uniform subpower bound B_{ℓ,D}(x,N)≪(DNx)^ε in the Friedlander–Iwaniec conjecture fails for A(s)=B(s)=ζ(s)^m. The mechanism is endpoint resonance: choosing x so that the phase 2πm(nx)^{1/m}+φ_m is at a cosine maximum when n=N makes the phase vary by O(1) over a block of length H ≍ N^{1-(a+1)/m} preceding N, and all those terms contribute with the same sign.

Load-bearing premise

The counterexample applies only if ζ(s)^m, with its pole of order m at s=1 and boundary spectral parameters κ_j=0, is an admissible automorphic series in the original conjecture; the paper asserts this on the basis of the original framework, but if that framework secretly requires cuspidal L-functions with simple poles, the counterexample would lie outside its scope.

Editorial extensions

If this is right

  • The uniform subpower estimate in Friedlander–Iwaniec Conjecture 1 is false for the zeta-power datum; any proof would have to exclude such poles or restrict the class of series.
  • The counterexample satisfies N≤x for m≥6, so relaxing or enforcing that condition does not save the conjecture in the zeta-product case.
  • The proposed deduction of the omega-free error-term conjecture (Conjecture 2) from Conjecture 1 via N≍x^{(m-1)/2} collapses for these examples, though Conjecture 2 itself is not disproved.
  • Any sharp-cutoff sum with positive coefficients bounded below by a constant and a slowly varying nonlinear phase will exhibit the same resonance; the obstruction is not peculiar to zeta powers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The endpoint-sector lemma suggests that any L-function with a pole of order at least 4 at s=1 (or otherwise yielding coefficients bounded below) can be used to build counterexamples; one might test this for products of Dirichlet L-functions with multiple poles.
  • The conjecture may remain plausible for cuspidal automorphic forms whose coefficients are not bounded below by a positive constant (e.g., Hecke eigenvalues with sign changes); the lower bound d_m(n)≥1 is essential to the construction.
  • A quick numerical check for m=4, a=0.4, N=10^6 using the explicit formula (3.4) should show |B_4| growing like N^{1/40}, which would settle the matter concretely.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper claims to disprove the Friedlander–Iwaniec sharp dual-sum Conjecture 1 by constructing explicit power-sized values of the sharply truncated dual sum for A(s)=B(s)=ζ(s)^m, m≥4. The core is an elementary endpoint-sector lemma (Theorem 2.1): for any phase κ(nx)^ν with 0<ν<1 and coefficients bounded below, one can choose x_N ≍ N^a so that the terminal H_N ≍ N^{1−ν(a+1)} terms lie in a fixed arc where the periodic weight has positive real part, forcing a block contribution ≫ N^{1−β−ν(a+1)}. Specialized with ν=1/m, β=(m+1)/(2m), κ=2πm and c_n=d_m(n), this yields Theorem 1.2 and Corollary 1.3: for m≥4 and 0<a<(m−3)/2, the dual sum is ≫ N^{(m−3−2a)/(2m)} along a sequence, contradicting the conjectured uniform subpower bound. The paper also gives an N≤x counterexample for m≥6 and an N ≍ x^{(m−1)/2} counterexample for m≥5.

Significance. If the Friedlander–Iwaniec conjecture is correctly interpreted to include ζ(s)^m with pole order m≥4 and arbitrary N, the paper's result is a clean, explicit refutation of a published conjecture. The proof is transparent and elementary, with no fitted constants, machine-verifiable computations, and explicit sequences; it also explains why the trivial bound in (1.10) is the best possible in this case. The paper's self-imposed limitations (Remark 2.3, Section 1.5) show awareness of the boundaries of the argument. The main risk is external: the scope of the original conjecture is not quoted, and the counterexample relies on a specific reading of [6]. The m≥6, N≤x part of Corollary 1.3(ii) is robust to the N≤x restriction.

major comments (2)
  1. [Section 1.3, Conjecture 1.1] The conjecture is presented as a paraphrase without quoting the original. The paper's claims that [6, pp. 494–495] explicitly allow products of zeta with poles of arbitrary finite order, and that (1.9) is uniform in N without the restriction 1≤N≤x, are load-bearing. If the original conjecture restricts N≤x, the m=4,5 counterexamples (with a<1 so N ≫ x_N) are outside the conjecture's range, leaving only m≥6 with N≤x. If the class of "automorphic series" is narrower than the paper's reading (e.g., cuspidal factors with simple poles), ζ(s)^m may not be a permissible datum. Please quote the exact statement of [6, Conjecture 1] and the cited passage, and amend the abstract/theorems if any restriction applies.
  2. [Section 3.1, Proposition 3.1] The verification that ζ(s)^m lies in the Friedlander–Iwaniec framework relies on the assertion "Re κ_j=0 is explicitly permitted in [6, (1.4)]" and "pole at s=1 of arbitrary finite order [6, pp. 494–495]." These are not demonstrated in the manuscript. Because the counterexample's validity depends entirely on this inclusion, the authors should reproduce the relevant conditions verbatim and confirm that the boundary case κ_j=0 and pole order m are allowed. If they are not, the counterexample does not lie in the class covered by the conjecture.
minor comments (4)
  1. [Section 1.6] Typo: "Organiztion" should be "Organization."
  2. [Abstract] Grammar: "Let a(n) and b(n) are arithmetic sequences" should be "Let a(n) and b(n) be arithmetic sequences."
  3. [Section 1.3] The condition "1≤N≤x" appears in (1.8) but is omitted in Conjecture 1.1; clarify whether the conjecture is stated for all N or for the same range, and ensure the notation B_{\ell,D}(x,N) is consistent.
  4. [Remark 2.3] The statement "This is only an amplitude-level statement..." is useful, but it might be placed earlier to avoid confusion about the role of Clozel–Sarnak's framework.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the endpoint construction is self-contained and the conjecture is the target, not an input.

full rationale

The derivation chain is self-contained. Theorem 2.1 is an elementary endpoint-sector lemma proved directly from a nonvanishing periodic profile, a positive coefficient lower bound, and a first-derivative scale; it does not presuppose the conjecture. Theorem 1.2 applies it to ζ(s)^m by way of Proposition 3.1, which verifies the Friedlander–Iwaniec normalization using only the classical zeta functional equation, the coefficient bound d_m(n) ≪ n^ε, and the quoted conditions (3.1)–(3.2). No fitted parameter is renamed as a prediction: the resonant x_N is explicitly constructed so that the terminal phase is exactly a cosine maximum, and the lower bound (1.16) measures the coherent terminal block. Conjecture 1.1 is the target of the refutation; the contradiction argument assumes it momentarily and derives its failure, which is a standard refutation rather than a circular use of the conjecture as an input. There are no self-citations by this author. The only load-bearing external dependency is the scope of the Friedlander–Iwaniec conjecture as actually stated in [6] — whether ζ(s)^m with pole order m ≥ 4 is an admissible automorphic series and whether the bound is required for N > x. That is a factual or interpretational correctness risk, and the paper itself flags related limitations in Remark 2.3 and Section 1.5, but it is not circular reasoning. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No new particles, mediators, or ledger entries are introduced; the only construction is a choice of x_N and H_N. No numerical data are fitted: the parameters a and λ are existential and range over all admissible values, so they are not listed as free parameters. The central claim rests on standard zeta facts, on the scope of the Friedlander–Iwaniec framework, and on the single-cosine form of the archimedean factor.

assumptions (3)
  • standard math ζ(s) has meromorphic continuation to C with a unique simple pole at s=1 and satisfies ζ(1-s)=π^{1/2-s}Γ(s/2)/Γ((1-s)/2)ζ(s); hence ζ(s)^m has a pole of order m and degree-m functional equation.
    Used in Proposition 3.1 to verify the Friedlander–Iwaniec datum; cited to Titchmarsh [27, Theorem 2.1].
  • domain assumption The Friedlander–Iwaniec framework admits products of zeta and Dirichlet L-functions and allows poles of arbitrary finite order at s=1, with boundary spectral parameters Re κ_j=0 permitted.
    Load-bearing: it puts ζ(s)^m inside Conjecture 1.1. The paper cites [6, pp. 494–495] for this and [6, (1.4)] for the boundary case. A skeptical reader should verify this against the original paper.
  • domain assumption For a real archimedean parameter array, the two exponential height modes in the Friedlander–Iwaniec summation formula combine into the single cosine in (1.7) with ℓ=m-3.
    Invoked in Proposition 3.1 and Remark 2.3 to identify B_m with Friedlander–Iwaniec's B_{ℓ,D}; the explicit Stirling-level computation is not carried out in the paper.

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Pith. "Pith review of Counterexamples of Friedlander--Iwaniec dual sums conjecture." pith.science (2026). https://pith.science/paper/PCF2GGCY

@misc{pith2026260716695,
  author       = {Pith},
  title        = {Pith review of: Counterexamples of Friedlander--Iwaniec dual sums conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCF2GGCY}},
  note         = {Machine review of arXiv:2607.16695}
}
abstract

Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\sum_{n\ge1}a(n)n^{-s}, \qquad B(s)=\sum_{n\ge1}b(n)n^{-s},$$ be the two Dirichlet series related by a certain functional equation. Let $m$ be the \emph{analytic degree} of the functional equation. For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\mathcal B_{\ell,D}(x,N) := \sum_{\substack{n\in\mathbb N\\ n\le N}} b(n)n^{-\beta_m} \cos\left( 2\pi m\left(\frac{nx}{D}\right)^{1/m} +\frac{\pi\ell}{4} \right),$$ where $D\ge1$ is the conductor, $\beta_m:=\frac{m+1}{2m}$, and $\ell=m-3-2k$ is determined by the archimedean weight $k$ of the functional equation. Their Conjecture 1 predicts that, for every $\varepsilon>0$, $$\mathcal B_{\ell,D}(x,N) \ll_{\varepsilon,\boldsymbol\kappa} (DNx)^\varepsilon,$$ uniformly in the variables $x$ and $N$, with the degree, conductor, and archimedean datum fixed. We give counterexamples to this prediction with $$A(s)=B(s)=\zeta(s)^m,\; m\geq 4$$ where $$\zeta(s):=\sum_{n\ge1}n^{-s} \qquad(\operatorname{Re}s>1)$$ is the Riemann zeta function.

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