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Zeros of the Dirichlet series of even zeta values

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Pith's one-line read The zero set of D(s)=Σ ζ(2n)n^{-s} is completely and unconditionally described: zero-free half-plane, a single real zero, Riemann-class strip zeros, and two simple left-half-plane strings.

desk verdict A careful, genuinely new zero classification for a non-Euler-product Dirichlet series; the main theorems hold up, and only minor tidying is needed. read the letter →

arxiv 2607.20758 v1 pith:IIIBQRZT submitted 2026-07-22 math.NT

William D. Banks This is my paper · ORCID
classification math.NT MSC 11M0611M2611M41
keywords DirichletseriesRiemannzetafunctionfunctionalequationzerosofa-pointsLipschitzsummationformulapolylogarithmHecke-type
topics The Riemann Hypothesis
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 gives a complete unconditional description of the zero set of the Dirichlet series $D(s)=\sum_{n\ge1} \zeta(2n) n^{-s}$, a series with positive real coefficients that has neither an Euler product nor a self-dual functional equation. Meromorphic continuation is immediate from $D=\zeta+E$ with E entire, but locating the zeros is not; the paper proves an exact functional equation of Hecke type, $D(s)=\Gamma(1-s)Z(1-s)$, whose dual series runs over complex logarithms of perfect squares, with Riemann's functional equation appearing as one column. That identity yields four families of zeros: a zero-free right half-plane, one real zero $\rho_0=0.2004\ldots$, strip zeros that are perturbed a-points of ζ and obey a Riemann–von Mangoldt law, and two complex-conjugate strings of simple left-plane zeros with explicit asymptotics. A sympathetic reader should care because complete zero-set classifications are extremely rare for Dirichlet series lacking Euler products and self-duality, and the proof is entirely unconditional.

What carries the argument

The key object is the exact functional equation of Hecke type (Theorem 3.1), obtained by applying the Lipschitz summation formula to each summand in $D(s)=\sum_k \mathrm{Li}_s(k^{-2})$ and summing the identities over k. For $\sigma<0$ it reads $D(s)=\Gamma(1-s)Z(1-s)$, with $Z(w)=\sum_{\omega\in\Omega} \omega^{-w}$ over the frequency set $\Omega=\{2\log k + 2\pi i \ell : k\in\mathbb{N}, \ell\in\mathbb{Z}\} \setminus \{0\}$, grouped by k because the double series is not absolutely convergent. In the left half-plane the proof reduces Z to its two leading frequencies — $\omega_\flat=2\log 2$ contributed by the entire part $E(s)=\sum(\zeta(2n)-1)n^{-s}$ and $\pm \omega_\sharp = \pm 2\pi i$ contributed by ζ itself — via a two-term approximation $Z(w)=\omega_\flat^{-w}+\omega_\sharp^{-w}+O(e^{-\eta u}(|\omega_\flat^{-w}|+|\omega_\sharp^{-w}|))$. The constant $L=\log(\omega_\sharp/\omega_\flat)=\log(\pi/\log 2)+i\pi/$

What would settle it

Compute $D(s)$ to, say, 40 digits at a point with $\sigma<0$ (e.g., $s=-5+10i$) using both the original Dirichlet series with the decomposition $D=\zeta+E$ and the right-hand side $\Gamma(1-s)Z(1-s)$ of the functional equation, truncating Z with a verified bound on the tail; agreement to the stated precision would support the identity, and any mismatch would refute the paper's central premise. Alternatively, search the left half-plane for a zero that violates the string asymptotic (5.2) or the spacing $2\pi/|L|$.

Watch

Extended reading notes

Core claim

The central claim is that the zero set of $D(s)=\sum \zeta(2n) n^{-s}$ is completely and unconditionally described. Using the Lipschitz summation formula termwise in the polylogarithm decomposition, the paper establishes an exact functional equation $D(s)=\Gamma(1-s)Z(1-s)$ for $\sigma<0$, where the dual series $Z(w)=\sum_{\omega\in\Omega} \omega^{-w}$ runs over the set $\Omega=\{\log k^2 + 2\pi i \ell : k\in\mathbb{N}, \ell\in\mathbb{Z}\} \setminus \{0\}$ with a mandatory grouping of terms; the k=1 column is precisely Riemann's functional equation. Interference between the two smallest frequencies, $\omega_\flat=2\log 2$ from the entire part E and $\pm \omega_\sharp = \pm 2\pi i$ from ζ, controls every zero of large modulus in the left half-plane. The paper proves that D is zero-free for $\sigma \ge \sigma_0 = 1.500127440\ldots$, that D has a r

Load-bearing premise

The entire zero analysis rests on the classical Lipschitz summation formula being applied termwise to the polylogarithm sum and the resulting identities being summed over k; if that functional equation $D(s)=\Gamma(1-s)Z(1-s)$ were invalid for $\sigma<0$, every subsequent zero-location argument collapses.

Editorial extensions

If this is right

  • For the series D, the full zero set is known without any unproved hypothesis: zero-free right half-plane, one real zero, a Riemann–von Mangoldt strip count, and two simple left-plane strings.
  • The strip zeros of D mirror the a-points of ζ for a near −(ζ(2)−1), so the value-distribution theory of ζ transfers to D at this quantitative level.
  • Riemann's functional equation appearing as a single column of the dual series suggests a new structural family of Hecke-type equations generated by special values of ζ.
  • The counting law N_D(T) agrees to leading order with that of ζ itself, despite D having neither Euler product nor self-duality, so zero counting alone does not detect those properties.
  • The exponentially decaying error in the string asymptotics means the left-plane zeros are eventually governed by a finite two-term exponential-sum model, making them computable to high precision.

Reading between the lines

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

  • The two-frequency interference picture should extend to the family D_α(s)=Σ ζ(αn)n^{-s} for real α>1: for α<2π/log 2 the same strings are predicted (the paper sketches this in Section 8); if correct, the entire one-parameter family would have a uniform zero-set description.
  • The author leaves open whether almost all strip zeros cluster near σ=1/2 (a Levinson-type statement); the bounded, nearly constant perturbation −E(s) makes a proof plausible by adapting mean-value estimates for ζ.
  • If the real zero ρ0 is indeed unique, a rigorous monotonicity proof of D on (0,1) would follow from a moderate amount of numerical verification of E′(σ) bounds; this is a concrete testable extension.
  • The method locates the left-plane zeros by Rouché's theorem around the two-term approximation; one can test the sharp constant in the error term by computing the distance from actual zeros to the predicted points for m up to a few hundred.
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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

0 major / 6 minor

Summary. The paper studies the Dirichlet series D(s)=Σ_{n≥1} ζ(2n)n^{-s}, which continues meromorphically to C with a single simple pole at s=1. The central input is Theorem 3.1, an exact Hecke-type functional equation D(s)=Γ(1−s)Z(1−s) for σ<0, where Z(w) is a dual series over the complex logarithms of perfect squares, obtained from the Lipschitz summation formula with a mandatory grouping of terms (Remark 3.2). Using this functional equation, the author proves: zero-free half-plane σ≥σ0=1.500127440… (Prop. 4.1); a real zero ρ0=0.200411339… with D(σ)>0 for σ≤0 and σ>1, uniqueness conjectural (Prop. 4.2, Conj. 4.3); finiteness of zeros in every fixed left strip (Prop. 4.4); in the far left half-plane, all zeros are simple and form two conjugate strings satisfying ρ_m = 1 − iπ(2m+1)/L + O(e^{-κm}) with explicit L, Θ, κ (Theorem 5.1); and in the region −1/4≤σ≤σ0, zeros are a-points of ζ near a0=−(ζ(2)−1), with counting law N_D(T)=T/(2π)log(T/(2π))−T/(2π)+O(logT) (Theorem 6.1). No Riemann-type hypothesis is used.

Significance. If accepted, this is a rare complete asymptotic zero classification for a Dirichlet series that has neither an Euler product nor a self-dual functional equation. The functional equation is exact and elegant, and Riemann's functional equation appears as a single column of the dual series. The zero analysis is genuinely parameter-free: all constants σ0, L, Θ, η, κ are computed from definitions rather than fitted. The string theorem is quantitatively sharp, with geometrically decaying error, and the counting law is derived by a clean argument-principle argument. The numerical section is unusually thorough: argument-principle counts, Newton iteration, explicit working precision, and an independent mpmath recomputation. These strengths make the paper a valuable test case for the broader problem of zeros of non-Euler-product Dirichlet series.

minor comments (6)
  1. [§4, Proposition 4.2] The proof uses the bound |ζ(−x)|≤1/2 on [0,1.3] without proof or citation. This is a real numerical input to the claim D(σ)>0 for σ≤0. Please supply a short justification (e.g., via the functional equation and monotonicity on that interval) or an explicit certified computation.
  2. [§4, Proposition 4.1] In the equality case, the sentence 'forces the phases n^{-it} to have the same value for all n≥2... occurs only if t=0' relies on the rational independence of log 2 and log(3/2). State or cite this elementary fact so the boundary case σ=σ0 is fully justified.
  3. [§6] The phrase 'critical strip' is used for the region −1/4≤σ≤σ0, which extends beyond the usual critical strip 0<σ<1 and also below 0. The definition in Theorem 6.1 is explicit, but the terminology may mislead; consider calling it the 'central rectangle' or 'strip region'.
  4. [§5, Theorem 5.1] The symbol L is used both for the complex constant L=log(π/log2)+iπ/2 and for its modulus |L|. This is notationally risky (e.g., in (5.1)–(5.2)); a separate symbol for the modulus would improve readability.
  5. [§2 and §8] The claim of a 'complete unconditional description' is slightly stronger than what is proved, since uniqueness of the real zero is conjectural (Conjecture 4.3) and Levinson-type clustering is left open (Section 8). The abstract and Section 8 do qualify these points; a sentence in the introduction would further align the language with the proven results.
  6. [§7] The certified notebooks are 'available from the author on request'. For reproducibility, it would be preferable to make the code and printed output publicly available in a permanent repository.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the functional equation comes from the classical Lipschitz/Riemann inputs, all constants are explicitly computed, and the numerics are checks, not fitted inputs.

full rationale

The paper's derivation chain is self-contained and non-circular. Theorem 3.1 obtains D(s)=Γ(1−s)Z(1−s) by applying the classical Lipschitz/Hurwitz formula (3.1) to the polylogarithm decomposition (1.3) and summing termwise under the k-grouping justified in Remark 3.2; the k=1 column is the standard Riemann functional equation (3.2), an external classical input, not a consequence of the paper's results. No parameter is fitted: σ0 is the unique root of the explicitly defined decreasing function h(σ), ρ0 is located by real-variable sign changes, and L, Θ, η, κ are defined via explicit quantities such as A=log(π/log 2) and |ω_{2,1}|/2π, with the admissibility of η proven in Lemma 5.4 by explicit inequalities. The strip-zero count in Theorem 6.1 is a standard argument-principle estimate using classical asymptotics for X(s) and ζ; the a-point interpretation is an immediate reformulation of ζ(ρ)=−E(ρ) with the quantitative bound (6.1), not an assumed conclusion. Tables 2 and 3 verify the proved asymptotics; the computations are not used as hypotheses. There are no load-bearing self-citations (indeed no self-citations at all), and no uniqueness theorem is imported from the author's prior work. The stated limitations—the left-open Levinson-type clustering question and the unrefined sector exponent—are explicit scope restrictions, not circular steps.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on classical analytic number theory (Lipschitz summation, Riemann's functional equation, Rouché–Jensen methods) rather than on new postulated entities. All constants are defined deterministically; no parameter is fitted to the zero data. The only unproved auxiliary assertion is the bound |ζ(−x)|≤1/2 on a real interval, which is peripheral to the two main theorems.

assumptions (7)
  • standard math Lipschitz summation formula (Hurwitz's formula): Li_s(e^{−λ}) = Γ(1−s) Σ_{ℓ∈Z}(λ+2πiℓ)^{s−1} for σ<0.
    Stated as (3.1) and used to derive the functional equation (3.3); the entire dual-series analysis depends on it. Classical, cited to [3,8,9,5].
  • standard math Riemann's functional equation ζ(s)=X(s)ζ(1−s) in the asymmetric form (3.2).
    Appears as the k=1 column of the dual series and is used in Lemmas 5.3 and in Section 6; standard input.
  • standard math Rouché's theorem, the argument principle, and Jensen's formula.
    Used in Theorem 5.1 (zero location and simplicity) and Theorem 6.1 (counting via argument principle).
  • standard math Stirling's formula and Γ(x+1) ≥ √(2πx)(x/e)^x.
    Used in Proposition 4.2 and in Lemma 5.4 for tail estimates.
  • standard math Rational independence of the logarithms of integers (unique factorization).
    Proposition 4.1 asserts the phases n^{−it} cannot be equal for all n≥2 unless t=0; correct by unique factorization but stated without proof.
  • domain assumption |ζ(−x)| ≤ 1/2 for 0 ≤ x ≤ 1.3.
    Asserted without proof in Proposition 4.2 to conclude D(−x)>0; numerically true and standard, but not derived in the text.
  • standard math Standard zeta estimates: |ζ(σ+it)| ≪ |t|^3 in fixed strips, asymptotic for X(σ+it), and 1/ζ(1+it) ≪ (log t)^7.
    Cited to Titchmarsh [12]; used in Propositions 4.2/4.4, Theorem 5.1 Step 4, and Theorem 6.1.

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Cite this review

Pith. "Pith review of Zeros of the Dirichlet series of even zeta values." pith.science (2026). https://pith.science/paper/IIIBQRZT

@misc{pith2026260720758,
  author       = {Pith},
  title        = {Pith review of: Zeros of the Dirichlet series of even zeta values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIIBQRZT}},
  note         = {Machine review of arXiv:2607.20758}
}
abstract

We give a complete unconditional description of the zero set of the Dirichlet series $D(s) := \sum_{n\ge1} \zeta(2n)\, n^{-s}$, which continues meromorphically to $\mathbb{C}$ with a single simple pole at $s=1$. The series possesses neither an Euler product nor a self-dual functional equation, and descriptions with this level of completeness are exceedingly rare for such series. The key input is an exact functional equation of Hecke type, obtained from the Lipschitz summation formula, which expresses $D$ in the left half-plane as a gamma factor times a dual series over the complex logarithms of the perfect squares; Riemann's functional equation appears as a single column of the dual series. The zeros fall into four families. The half-plane $\sigma \ge \sigma_0 = 1.5001\cdots$ is zero-free, and $D$ has a real zero $\rho_0 = 0.2004\cdots$, conjecturally its only one. The zeros in the critical strip are perturbed $a$-points of $\zeta$ for values of $a$ near $-(\zeta(2)-1)$, and their counting function obeys a Riemann-von Mangoldt law. The remaining zeros form two complex-conjugate strings that recede into the left half-plane along explicit rays, are eventually simple, and satisfy an asymptotic with geometrically decaying error. The string geometry is governed by interference between the two smallest frequencies of the dual series, $2\log 2$ contributed by the entire part of $D$ and $\pm 2\pi i$ contributed by $\zeta$. No hypothesis of Riemann type is assumed at any point.

Figures

Figures reproduced from arXiv: 2607.20758 by the authors.

Figure 1
Figure 1. Computed zeros of D in the window |t| ⩽ 50. The two strings of left-plane zeros recede along the dashed rays arg(1−s) = ±Θ. The strip zeros shadow a-points of ζ. The star marks the real zero ρ0 = 0.2004 · · · , and the cross marks the pole at s = 1. The shaded half-plane σ ⩾ σ0 is zero-free. Tables 2 and 3 extend beyond the plotted range [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. D(σ) on the real segment (−0.1, 0.9). The sign change occurs at ρ0 = 0.2004 · · · , numerically the only real zero of D. It remains to prove that D(−x) > 0 for all x ⩾ 0. Since ζ(2n) − 1 ⩾ 4 −n for every n ∈ N, we have E(−x) ⩾ X m∈N mx 4 −m = Li−x [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The two dominance regions for Z(w) in the quadrant u > 0, v ⩾ 0. Below the ε-cone about the ray arg w = Θ, the term ω −w ♭ dominates. Above the cone, the term ω −w ♯ dominates. The circles are the points wm defined in the proof of Theorem 5.1. They lie on the ray, where neither term dominates. The zeros ρm of (5.2) correspond to these points under s = 1 − w. For every ω ∈ Ω, we have [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The counting rectangle in the proof of Theorem 6.1, drawn for t0 = 2 and T = 40. The squares mark the strip zeros of D with t < 40, listed in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

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Reference graph

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