Note on the $a$-points of the Riemann zeta function

Min-Jie Luo; Peng-Cheng Hang

REVIEW 2 major objections 5 minor 16 references

Note on the $a$-points of the Riemann zeta function

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves a shifted, weighted mean-value formula for the a-points of the zeta function, with explicit arithmetic correction terms.

desk verdict Useful δ-shifted a-point mean value formula for a≠1, but Theorem 2.3 is ill-posed at a=1 and needs fixing before it is complete. read the letter →

arxiv 2411.13255 v2 pith:ONSK3KDK submitted 2024-11-20 math.NT

Peng-Cheng Hang , Min-Jie Luo This is my paper · ORCID

classification math.NT MSC 11M0611M2641A60

keywords Riemannzetafunctiona-pointsvaluedistributiondiscretemeantheoremhigherderivativespartialfractionexpansionvonMangoldtMöbius

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 establishes an asymptotic formula for sums of the derivative of the zeta function over its $a$-points, the points $\rho_a$ where $\zeta(\rho_a)=a$, shifted by a small imaginary amount $\delta$ and weighted by $X^{\rho_a}$. The central result, Theorem 2.3, expresses such a sum as the corresponding sum over ordinary zeros minus an explicit arithmetic correction built from the divisor sum $\sum_{mr=X}(\Lambda(r)+c_a(r))m^{-i\delta}\log m$, minus an $a$-dependent $K$-term, plus an error of size $O(T^{1/2}\log^7 T)$, uniformly in the shift parameter $\alpha$. This corrects a previously published formula, which omitted the $c_a(r)$-type and $a$-dependent contributions. The result matters because it supplies a $\delta$-shifted, $X$-weighted mean-value theorem for $a$-points, and differentiating the formula in $\alpha$ yields explicit expansions for sums of higher derivatives of $\zeta$ at $a$-points, extending known results for ordinary zeros.

What carries the argument

The load-bearing tool is the partial-fraction expansion $$\frac{\zeta'(s)}{\zeta(s)-a}=\sum_{|t-\gamma_a|\le1}\frac{1}{s-\rho_a}+O(\log(|t|+1)),\qquad -1\le\$\sigma$\le2,$$ which represents the ratio as a sum of simple pole terms over nearby $a$-points. This expansion, quoted from earlier work and here buttressed by a reformulated description of the trivial $a$-points via a zero-counting argument, allows the proof to turn the $a$-point sum into a contour integral of $\frac{\zeta'(s)}{\zeta(s)-a}\zeta'(s+i\delta)X^s$ around a rectangle. The right vertical side is evaluated by expanding the ratio into the Dirichlet series $\sum_{r\ge2}c_a(r)r^{-s}$; the left vertical side uses the expansion of $1/(\zeta(s)-a)$ in powers of $a/\zeta(s)$; and the horizontal sides are controlled by the partial-fraction bound. The coefficients $c_a(r)$ and the auxiliary function $K_\delta^{(1)}$ carry the arithmetic content of the correction terms.

What would settle it

Take a concrete non-integer, non-reciprocal case such as $a=1$, $X=\sqrt2$, $\alpha=1$, $\tau=2$, and compute both sides of (2.4) numerically at $T=10^5$ by locating the $a$-points of $\zeta(s)=1$ in the rectangle. Since $\Delta(\sqrt2)=\Delta(1/\sqrt2)=0$, the theorem predicts that the $a$-point sum equals the ordinary zero sum within $O(T^{1/2}\log^7 T)$; a difference with a term of size comparable to $T$ would disprove the formula or the partial-fraction expansion behind it.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 2.3: for fixed $a\in\mathbb{C}$, $X>0$, and $\tau\ge|\delta|+1$, with $0\ne\delta=2\pi\alpha/\log(T/(2\pi X))\ll 1$, as $T\to\infty$, $$\sum_{\tau<\gamma_a\le T}\zeta'(\rho_a+i\delta)$X^{{\rho_a}}$ = \sum_{\tau<\gamma\le T}\zeta'(\rho+i\delta)$X^{{\rho}}$ - \$\Delta$(X)\frac{T}{2\pi}\sum_{mr=X}(\Lambda(r)+c_a(r))$m^{{-i\delta}}$\log m - aK_\$delta^{{(1)}}$\left(\frac{T}{2\pi}\right) + O\left($T^{{1/2}}$\$log^{7}$ T\right),$$ uniformly in $\alpha$. Here $\Lambda$ is the von Mangoldt function, the coefficients $c_a(r)$ are defined by the Dirichlet series $\zeta'(s)/(\zeta(s)-a)=\sum_{r\ge2}c_a(r)r^{-s}$, and $K_\delta^{(1)}$ is an explicit combination of Möbius- and von Mangoldt-weighted sums over divisors of $1/X$. In words: the average of $\zeta'$ at shifted $a$-points differs from the same average at ordinary zeros only through arithmetic terms supported on the divisor pair $mr=X$, plus a term that vanishes unless $1/X$ is an integer. The paper also derives the integer-$X$ specialization, the $\delta\to0$ limit, and the $a=0$ case, recovering and correcting earlier results.

Load-bearing premise

The proof depends on a formula that rewrites the logarithmic-derivative-like ratio of the zeta function near each $a$-point as a sum of simple pole terms plus a controlled error; if that formula fails for some $a$, the main result collapses.

Editorial extensions

If this is right

Differentiating (2.4) with respect to $\alpha$ yields explicit asymptotic expansions for sums of higher derivatives $\zeta^{(n)}(\rho_a)$ at $a$-points, as stated in Theorem 8.1; setting $a=0$ recovers known higher-derivative mean-value formulas for zeros.
For positive integer $X$, Corollary 2.4 corrects the published evaluation of $\sum_{1<\gamma_a\le T}\zeta'(\rho_a)X^{\rho_a}$, adding the previously missing $c_a(r)$ and $a$-dependent terms.
The uniformity in $\alpha$ means the formula holds simultaneously for every allowed shift $\delta$, so it can be differentiated or integrated in $\alpha$ to generate families of related mean values.
The factors $\Delta(X)$ and $\Delta(X^{-1})$ show that the asymptotic takes different shapes according to whether $X$, $1/X$, or neither is an integer; for a generic non-integer $X$, the $a$-point sum equals the zero sum up to the stated error.

Reading between the lines

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

A clean numerical check of the $X=\sqrt2$, $a=1$ case would isolate the theorem's strongest new prediction: because both $\Delta$-terms vanish, the $a$-point and zero sums should agree to leading order, testing the partial-fraction expansion without needing the $c_a(r)$ coefficients.
Repeating the contour proof with $\zeta^{(n+1)}$ in the integrand should give explicit formulas for $\sum \zeta^{(n)}(\rho_a+i\delta)X^{\rho_a}$ for all $n$, with the same arithmetic corrections expressed through higher von Mangoldt functions.
Letting $a$ tend to $0$ with $T$ should produce a transition formula interpolating between the $a$-point and zero sums; the shape of that interpolation may reveal how trivial $a$-points relate to the usual trivial zeros.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does something genuinely useful: it extends Fujii's discrete mean value formula to a δ-shifted version for ordinary zeros (Theorem 2.1), and it gives the first δ-shifted weighted mean-value formula for a-points (Theorem 2.3), correcting a mistake in Jakhlouti-Mazhouda's (1.5). The contour-integral proof is detailed and, for a ≠ 1, the main line of argument is plausible. The authors also carefully reformulate Garunkštis–Steuding's results on trivial a-points; Lemma 4.8 is cited from that work but at least they provide the context. I give credit for the honesty about error terms and the explicit removal of the |T−γ| restriction.

The soft spots are real, and one is load-bearing. The stress-test note is right: Theorem 2.3 is stated for all a ∈ C, but the coefficients c_a(r) defined by (2.5) do not exist at a = 1. Let σ→∞; the left-hand side of (2.5) tends to −log 2, while a Dirichlet series with c_1(1) = 0 tends to 0. So (2.5) cannot define c_1(r). The proof explicitly assumes a ≠ 1, and the one-sentence workaround with f(s) = 2^s(ζ(s)−1) does not supply a definition for the c_1(r) appearing in the statement of the theorem. Either the theorem should be restricted to a ≠ 1, or the authors need to define c_1(r) through a limiting procedure or a different expansion. As written, the a = 1 case is unsupported.

Two smaller issues. First, Theorem 8.1 is stated without proof; the text says repeating the proof of Theorem 2.3 would give it, but that is not a proof in the paper. Second, Lemma 4.8 (the partial fraction expansion) is taken from Garunkštis–Steuding and depends on delicate lower bounds for |ζ(s)| in the left half-plane; the authors reformulate the key lemmas but only sketch the proof. If that decomposition fails for some a, the bound on the horizontal integral S_T and hence the main term in Theorem 2.3 could be incomplete. This is a known difficult part and deserves a careful check. The choice of τ avoiding a-points on the boundary is also asserted without full justification, but that is a minor fix.

Who is this for? Analytic number theorists working on discrete mean values of ζ and its a-points. The paper deserves a serious referee, not a desk reject. My recommendation: send it to review, but the referee should insist that the a = 1 case be fixed or excluded, and should verify the Garunkštis–Steuding input. With that, the a ≠ 1 results are likely to stand.

Referee Report

2 major / 5 minor

Summary. The paper studies a-points of the Riemann zeta function, i.e., solutions of ζ(s)=a. The main result (Theorem 2.3) is an asymptotic formula for the weighted sum S_T(a,δ)=Σ_{τ<γ_a≤T} ζ'(ρ_a+iδ) X^{ρ_a}, valid uniformly in α where δ=2πα/log(T/(2πX)). The formula expresses S_T in terms of the analogous sum over ordinary zeros, a correction term involving the arithmetic coefficients c_a(r), and a further term K_δ. The paper also proves a δ-shifted version of Fujii's mean-value formula for ordinary zeros (Theorem 2.1), derives a corollary for integer X (Corollary 2.2), and obtains a corrected version of a result of Jakhlouti and Mazhouda (Corollary 2.4). The proofs use contour integration over rectangles, the partial fraction expansion of ζ'(s)/(ζ(s)-a), and known estimates for ζ(s).

Significance. If correct, Theorem 2.3 is a new weighted, δ-shifted discrete mean-value formula for a-points, and it corrects an earlier formula (1.5). It also reveals a dependence on X through Δ(X) and Δ(X^{-1}), giving a richer structure than previous results. The computations are detailed, the error terms are explicit, and the paper carefully restates known results on the distribution of trivial a-points. However, the a=1 case of the main theorem is not treated correctly, and the proof has a gap concerning the choice of τ; these issues must be resolved before the main claim can be accepted.

major comments (2)

[§7, Eq. (2.5) and Theorem 2.3] The defining equation for c_a(r) fails at a=1. Since ζ'(σ)/(ζ(σ)-1) tends to -log 2 as σ→∞ along the real axis, whereas a Dirichlet series with c_1(1)=0 tends to 0, no coefficients satisfying (2.5) with c_a(1)=0 exist for a=1. The proof of Theorem 2.3 explicitly assumes a≠1 after a one-sentence note on f(s)=2^s(ζ(s)-1), and the subsequent derivation of S_L, S_R, and the final formula is only for a≠1. Consequently, Theorem 2.3 as stated for all a∈C is ill-posed and unsupported; the statement should exclude a=1 or give a correct separate treatment.
[§7, after Eq. (7.1)] The text reads "we may take τ large enough such that there are no a-points on the boundary ∂R", but τ is fixed at the start of the theorem. The proof as written covers only τ chosen to avoid the (discrete) set of ordinates of a-points; either the theorem must include this condition on τ, or a limiting/indentation argument is needed for arbitrary fixed τ.

minor comments (5)

[§7.1, display after definition of a_k] In the formula for S_R, the subscript "Σ_{mr=k}" should be "Σ_{mr=X}"; the summation variable k is not defined in that context.
[§8, Theorem 8.1] Theorem 8.1 is stated without proof; either supply the promised "repeating the proof" details or clearly mark it as a remark/sketch, since unproved theorems cannot be part of the formal results.
[§4, Lemma 4.8] Lemma 4.8 is quoted from [4, p. 8] and is essential for the S_T estimate in Theorem 2.3; please include at least a sketch of the proof or an explicit verification of the O(log(|t|+1)) bound.
[Throughout] The notation "τ /greaterorequalslant|δ| + 1" is a rendering artifact; please ensure all inequalities are typeset correctly in the final version.
[§8, Remark 8.1] The remark that S_T(a,δ) is "analytic in δ" and, as a function of a, "analytic at a=0 and discontinuous elsewhere" is not substantiated; please clarify or remove.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main asymptotic formulas are derived by explicit contour integration from independent cited lemmas, with no fitted parameter renamed as a prediction.

full rationale

The derivation chain is self-contained in the relevant sense: Theorem 2.1 is proved by a contour integral over the rectangle using standard estimates for the xi-function, Perron-type integrals, and Fujii's known formulas; Theorem 2.3 is then obtained by a separate contour integral of ζ'(s)/(ζ(s)-a) ζ'(s+iδ)X^s, with the coefficients c_a(r) merely defined by the Dirichlet-series expansion (2.5) and used to evaluate the right-hand edge of the contour, not fitted to any data. The auxiliary lemma about the partial fraction expansion (Lemma 4.8) is quoted from Garunkštis and Steuding and is not a restatement of the paper's main claim. The reductions to known results at δ=0 (Fujii's formula) or a=0 (ordinary zeros) are consistency checks rather than circular dependencies. There is no self-citation chain carrying the argument: the cited works are by Fujii, Garunkštis–Steuding, Hughes–Pearce-Crump, and Steuding, and none of these is used to assume the very formula being proved. The skeptical concern about the case a=1 (where the defining expansion (2.5) fails since the left side tends to -log 2 while the right side would tend to 0) is a substantive correctness or well-posedness issue in the statement of Theorem 2.3, not a circularity: even if the theorem is unsupported at a=1, the derivation for a≠1 does not reduce to its inputs by definition. Accordingly, no circular step is identified and the score is 0.

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

The central claim rests on standard analytic number theory inputs (counting formulas, bounds for ζ and χ, Perron formula, partial fraction expansion) all from the cited literature. No free parameters are introduced; the constants C0, C1 and the coefficients c_a(r) are defined by standard series. No new entities are postulated. The main new ingredient is the contour computation itself, not a new axiom.

assumptions (7)

domain assumption Riemann-von Mangoldt-type counting formula N_a(T) = (T/2π) log(T/(2π e^{c_a})) + O(log T).
Quoted from Landau (1.2); used in (5.1) and Section 7 to control errors when removing the spacing condition and to count a-points on the contour.
standard math Standard bounds for ζ^{(n)}(σ+it) and ζ'/ζ on horizontal lines (Lemmas 4.1-4.3, 4.5), quoted from Hughes-Pearce-Crump and Pearce-Crump.
Used throughout to bound contour integrals in Sections 5 and 7.
domain assumption Existence and location of trivial a-points: only finitely many off the sequence near -2k and one near each -2k for large k (Lemmas 4.6-4.7).
Reformulated from Garunkštis-Steuding; needed for the partial fraction expansion and to justify the residue computation in Theorem 2.3.
domain assumption Partial fraction expansion ζ'(s)/(ζ(s)-a) = Σ_{|t-γ_a|≤1} 1/(s-ρ_a) + O(log(|t|+1)) (Lemma 4.8).
Quoted from Garunkštis-Steuding; used to bound the top-edge integral ST in Section 7.
standard math Truncated Perron formula (Montgomery-Vaughan, Corollary 5.3).
Used in Corollary 2.2 and Remark 8.1 to turn Dirichlet series into finite sums.
standard math Fujii's estimate Σ_{mr≤x} Λ(r) r^{iδ} = ζ(1+iδ)/(1+iδ) x^{1+iδ} - (ζ'/ζ)(1-iδ)x + E(x) (6.4).
Used in Corollary 2.2 to simplify the main term.
standard math Coefficient expansion ζ'(s)/(ζ(s)-a) = Σ_{r≥2} c_a(r)/r^s for ℜ(s)>1 (2.5), from Steuding.
Defines c_a(r) that appear in Theorem 2.3 and Corollary 2.4.

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Pith. "Pith review of Note on the $a$-points of the Riemann zeta function." pith.science (2026). https://pith.science/paper/ONSK3KDK

@misc{pith2026241113255,
  author       = {Pith},
  title        = {Pith review of: Note on the $a$-points of the Riemann zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ONSK3KDK}},
  note         = {Machine review of arXiv:2411.13255}
}

abstract

For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$ shown by Garunk\v{s}tis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,\delta)=\sum_{\tau<\gamma_a\leqslant T}\zeta'(\rho_a+i\delta)X^{\rho_a},\quad T\to\infty,\] where $0\ne\delta=\frac{2\pi\alpha}{\log\frac{T}{2\pi X}}\ll 1$, and $X>0$ and $\tau\geqslant|\delta|+1$ are fixed. We also find the interesting varied behavior of $S_T(a,\delta)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

H. Bohr, E. Landau and J. E. Littlewood, Sur la fonction ζ(s) dans le voisinage de la droite σ = 1 2 , Bull. Acad. Roy. Belg. (1913) 3–35

work page 1913

[2] [2]

Fujii, On a conjecture of Shanks, Proc

A. Fujii, On a conjecture of Shanks, Proc. Japan Acad. 70(4) (1994) 109–114

work page 1994

[3] [3]

Fujii, On the distribution of values of the derivative of the Riemann zeta function at its zeros, I, Proc

A. Fujii, On the distribution of values of the derivative of the Riemann zeta function at its zeros, I, Proc. Steklov Inst. Math. 276 (2012) 51–76

work page 2012

[4] [4]

Garunkˇ stis and J

R. Garunkˇ stis and J. Steuding, On the roots of the equati on ζ(s) = a, Abh. Math. Semin. Univ. Hamb. 84(1) (2014) 1–15

work page 2014

[5] [5]

S. M. Gonek, Mean values of the Riemann zeta-function and its derivatives, Invent. Math. 75 (1984) 123–141

work page 1984

[6] [6]

S. M. Gonek, A formula of Landau and mean values of ζ(s), in Topics in Analytic Number Theory (University of Texas Press, Austin, 1985), pp. 92–97

work page 1985

[7] [7]

Hughes and A

C. Hughes and A. Pearce-Crump, A discrete mean-value the orem for the higher derivatives of the Riemann zeta function, J. Number Theory 241 (2022) 142–164

work page 2022

[8] [8]

A. E. Ingham, Mean-value theorems in the theory of the Rie mann Zeta-function, Proc. Lond. Math. Soc. 27 (1927) 273–300

work page 1927