Infinite log-concavity of the Taylor coefficients of the Riemann xi-function

Jianxi Mao; Yanxin Liu

REVIEW

Infinite log-concavity of the Taylor coefficients of the Riemann xi-function

Yanxin Liu, Jianxi Mao

Combinatorics · Submitted 4 Oct 2026 · 29 references

Pith's verdict: Genuine result: unconditional strict infinite log-concavity of the xi-function coefficients, resolving Zhu's conjecture, with the real risk confined to the finite interval certificate rather than the analytic chain.

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arXiv ID2610.04972 (version 1)

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CitationJianxi Mao, Yanxin Liu. "Infinite log-concavity of the Taylor coefficients of the Riemann xi-function." arXiv:2610.04972 (2026). https://pith.science/paper/AWOGTOG6

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Claim paper

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Editor's note

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The headline is that this paper appears to be right, and it is a real result. Liu and Mao prove that the Taylor coefficients of the Riemann xi-function are strictly infinitely log-concave without assuming the Riemann hypothesis. That resolves Zhu's conjecture and turns Brändén's conditional implication into an unconditional theorem. The significance is genuine, not incremental.

I read the analytic chain carefully: Lemma 3.2's saddle-point estimate, the propagation to |g0 − Q| ≤ 20/n² in Lemma 3.3, the disk-nested iteration in Lemma 3.4, and the closure in Proposition 3.5. No internal error. The structure is honest about the two-index problem—coefficient index and iteration order—and solves it with the strong ratio threshold (3+√5)/2 and the finite τ(n), which is the right idea. The margins at the analytic/finite boundary are comfortable: ng0(n) ≥ 5/4 against a worst-case loss of 1/8 at n = 12000.

The load-bearing risk is exactly what the stress-test flags: Proposition 4.5's interval certificate for n ≤ 12000. The paper is admirably transparent here—code, SHA-256 hash, exact binary interval endpoints, outward rounding, a reproduction command. But the certificate depends on mpmath's exp, log, cos, pi, and sqrt evaluating with guaranteed directed rounding, and mpmath offers no formally verified guarantee for transcendental evaluation. A subtle rounding error could break an interval endpoint. The tightest ratio, about 10.904 at n = 12000, leaves a 9% margin over 10, so failure is unlikely but not impossible. This is a real gap, and it is the only one. The 4^-256 quadrature terms and the 10^-300 tail bounds are not where the risk sits.

Note the paper does not hide any of this: the appendix states the reliance on mpmath plainly, and the supplementary script check_analytic_constants.py verifies 34 scalar inequalities, though with the same primitives. So the right fix is an independent check, not a repair.

My recommendation: send it to a serious referee. The referee should audit the constants in Lemma 3.2 and, ideally, re-run the finite certificate with an independent interval engine (Arb, or verified crlibm-based rounding) rather than trusting mpmath. But the result is important, the proof is honestly structured, and I found no fatal flaw and no circularity. This deserves referee time, with the certificate cross-check as the one condition.

Referee report

Significance

If Theorem 1.1 is correct, it is a definitive resolution of a conjecture that was previously known only conditionally on RH; the unconditional strict infinite log-concavity (with the strong-ratio threshold (3+√5)/2 in place) is a substantial advance in the study of the xi-function's coefficient sequence and of iterated Turán inequalities. The paper's strengths are genuine: the analytic part is a self-contained derivation with explicitly stated hypotheses and explicit constants at every step (Lemma 2.2's disk-nesting conditions, Lemma 3.1's sector conditions, Lemma 3.4's induction with S_m < 1500/n²), the saddle-point deformation (contour pieces, Gaussian main term, tails, distant-left and vertical segments) is carried out with explicit error terms, and the closure margin ng₀(n) ≥ 5/4 against 1500/n ≤ 1/8 at n = 12000 is comfortable. The finite and analytic ranges meet consistently at n = 12000, where both give m = 14 (⌈log₂ 12000⌉ = 14). The paper ships the certificate code, a source hash, exact binary endpoint encodings, and reproduction instructions, which is commendable.

Major objections (2)

[Appendix A.2–A.3; Proposition 4.5, Eqs. (4.19)–(4.21), (4.29)–(4.30)] The soundness of the finite certificate rests on the claim that all interval operations use outward rounding (Appendix A.3). For the polynomial/arithmetic steps this is justified, but the enclosure endpoints for the transcendental evaluations are not formally certified. Concretely, exp and log appear in Q_n via K₁₆ in (4.19), and π, cos(1/4), exp, log appear in M_n in (4.20) and in the tail bound in (4.21). The text states that 'approximate mp calculations are used only to select candidate node brackets and intervals [ℓ_n, u_n] ... subsequently certified by outward interval inequalities', but those certifications themselves evaluate exp, log, cos, π (and the code also uses sqrt) by the same mpmath routines, and mpmath provides no formally verified directed-rounding guarantee for arbitrary-precision transcendental evaluation. Since Proposition 4.5 carries the entire range 1 ≤ n ≤ 12000, and the tightest reported ratio at n = 12000 ([10.904298..., 10.904298...]) has only a ≈9% margin over 10, a subtle endpoint error in any one transcendental evaluation could, in principle, break the r₁₄,n > 10 claim. I ask the authors to close this gap: either replace the mpmath evaluations with a library with formally proven enclosures (e.g., Arb-style interval arithmetic), or add an independent second implementation and cross-check all stored endpoints, or give an explicit, verifiable error analysis for each elementary function at the 130-digit working precision. The 9% margin is reassuring but does not, by itself, establish that the rounding is certified.
[Appendix A.3; Proposition 3.5 and the 34 scalar constants] The same issue propagates to the boundary between the analytic and finite ranges. The script scripts/check_analytic_constants.py is said to verify the 34 scalar inequalities used in the analytic estimates 'by outward interval arithmetic'; but if that verification uses the same unverified mpmath transcendental routines, then the analytic constants themselves (e.g., the (3.5)-(3.7) bounds and the numerical factors in Lemma 3.2's error terms) are only certified empirically. The analytic derivation in Section 3 is internally consistent with comfortable slack at the closure point ng₀(n) ≥ 5/4 with 1500/n ≤ 1/8, so I do not see an internal error there; but the paper should state precisely which correctness properties are being assumed of the interval-arithmetic backend on which both the finite certificate and the constant checks depend.

Minor objections (3)

[Section 4, Lemma 4.2(ii) / Lemma 4.4] In the proof of (4.7) the bound 4·17⁴·e^{−864} < 10^{−300} is asserted with the remark that it 'can also be verified directly by outward interval arithmetic'; for full rigor this is fine, but the sentence could give the actual verified bound or refer to the appendix script that certifies it.
[Section 5, proof of Theorem 1.1] The threshold τ(n) = 14 for 1 ≤ n < 12000 and the analytic threshold ⌈log₂ n⌉ coincide at n = 12000 (both equal 14); it would help the reader to see this explicitly stated, since the closure of the two ranges depends on it.
[Appendix A.1] The reproduction command and the exact JSON endpoint encodings (sign/mantissa/exponent/bitcount) are welcome; I would suggest adding a note that the decimal displays in Appendix A.2 (e.g., the n = 12000 ratio interval) are outward-rounded conversions, as is already stated, so that readers do not attempt to re-verify from the printed decimals alone.

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The paper

The paper claims

This paper proves that the sequence of even Taylor coefficients of the Riemann xi-function at its central point, λ_n = ξ^{(2n)}(1/2)/(2n)!, is strictly infinitely log-concave, and it does so without assuming the Riemann hypothesis.

The reading

The paper proves that the sequence of even Taylor coefficients of the Riemann xi-function at its central point, λ_n = $ξ^{{(2n)}}$(1/2)/(2n)!, is strictly infinitely log-concave, and it does so without assuming the Riemann hypothesis. Since RH is equivalent to F(x) = ξ(1/2 + √x) lying in the Laguerre–Pólya class, and membership in that class forces infinite log-concavity, the new content is that the property holds unconditionally, independent of where the zeros lie. The result settles Zhu's 2023 conjecture, which had previously been pushed only to 6-log-concavity.

What carries the argument

The load-bearing object is the logarithmic-ratio sequence g_{m,n} = log r_{m,n}, evolving under g_{m+1,n} = 2g_{m,n} − Δ²h(g_{m,n}), h(t) = log(1−$e^{{−t}}$), which turns each application of the Turán operator L into a convexity-preserving second-difference step. The initial datum comes from a saddle-point estimate of the integral H defining the ξ-coefficients: on |arg z| ≤ π/4, |z| ≥ 4000, |H/H* − 1| ≤ 30/T, where the saddle is the Lambert W point w = W(2z/π) and H* is an explicit closed form; this yields the leading term Q(z) = 2w/(z(1+w)) for g_0 with error 20/n². Iteration on nested disks keeps positivity, and the golden-ratio threshold (3+√5)/2 is the fixed point of r ↦ (r−1)², so crossing it once locks in infinite log-concavity for all higher iterates.

Extended reading notes

Core claim

The central claim is Theorem 1.1: for F(x) = ξ(1/2 + √x) = Σ λ_n x^n with λ_n = $ξ^{{(2n)}}$(1/2)/(2n)!, one has (L^m λ)_n > 0 for all m,n ≥ 0, so the sequence is strictly infinitely log-concave. The proof works with logarithmic ratios g_{m,n} = log r_{m,n}, where r_{m,n} = A²_{m,n}/(A_{m,n-1}A_{m,n+1}) and A = L^m λ; positivity of g is exactly log-concavity at iteration m. These ratios propagate by the recurrence g_{m+1,n} = 2g_{m,n} − Δ²h(g_{m,n}) with h(t) = log(1−$e^{{−t}}$). The initial ratios are estimated from an integral representation Λ(z) = H(z)/Γ(2z+1) for λ_n, using a saddle-point estimate with the Lambert W function as saddle: |H/H* − 1| ≤ 30/T on the sector |arg z| ≤ π/4, |z| ≥ 4000, which yields |g_0(z) − Q(z)| ≤ 20/n² with Q(z) = 2w/(z(1+w)), w = W(2z/π). The iterates are then controlled on successively smaller disks, and the ratio r_{τ(n),n} is shown to exceed the golden-ratio threshold (3+√5)/2 at a finite τ(n): at τ(n) = ⌈log₂n⌉ for n ≥ 12000 and at m = 14 for the finitely many n below 12000 (the latter by certified interval arithmetic). A closure lemma shows that once r ≥ (3+√5)/2 holds at one iteration, it persists forever, because (r−1)² ≥ r exactly in that range.

Load-bearing premise

The proof rests on the explicit saddle-point estimate |H(z)/H*(z) − 1| ≤ 30/T (T ≥ 630) and its propagated consequence |g_0(z) − Q(z)| ≤ 20/n² on the disk |z − n| ≤ n/2, together with the computer certification that r_{14,n} > 10 for n ≤ 12000; if any stated constant margin fails, or the interval-arithmetic certificate is wrong, the split between the analytic range (n ≥ 12000) and the finite range collapses.

Editorial extensions

If this is right

Every finite iterate of the Turán operator L applied to (λ_n) yields a positive sequence, so all finite-order Turán inequalities for the even Taylor coefficients of ξ at 1/2 hold unconditionally.
Zhu's 2023 conjecture is resolved: infinite log-concavity holds without RH, strengthening Brändén's conditional result into an unconditional statement.
The proof exhibits a reusable template — an analytic saddle estimate on an infinite range, a certified finite check on the remaining range, and a golden-ratio closure lemma — giving a practical route to infinite log-concavity for other moment-like sequences.
Infinite log-concavity is strictly weaker than RH, so this property is decoupled from the location of the zeros of ξ; it holds whether or not they line up on the critical line.

Reading between the lines

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

The closure lemma suggests a general criterion: any coefficient sequence that can be analytically interpolated and shown to satisfy r_{τ(n),n} > (3+√5)/2 at finitely many indices is infinitely log-concave; the same scheme could certify other sequences (partition numbers, other moment sequences) once a strong-ratio bound is available.
The saddle-point estimate |H/H* − 1| ≤ 30/T also gives an explicit zero-free region for H and quantitative control on the coefficients, a bonus tool that could be reused for other questions about ξ.
The golden-ratio threshold is probably not sharp; a smaller threshold would shorten the finite verification and could push the analytic/finite boundary down from n = 12000.

Original abstract

The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre--P\'olya class. Br\"and\'en [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre--P\'olya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\lambda_n)_{n\ge0}$. In this paper, we prove that the sequence $(\lambda_n)_{n\ge0}$ is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-06: 1 request, 5,529 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.80 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre--P\'olya class. Br\"and\'en [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre--P\'olya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\lambda_n)_{n\ge0}$. In this paper, we prove that the sequence $(\lambda_n)_{n\ge0}$ is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre--P\'olya class.	background 1.00	0.05	0.03	asserted 1.00
s1	Br\"and\'en [J. Reine Angew.	background 0.99	0.03	0.01	unclear 0.98
s2	Math., 2011] proved that if an entire function in the Laguerre--P\'olya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave.	background 1.00	0.03	0.03	asserted 1.00
s3	Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\lambda_n)_{n\ge0}$.	background 0.67	0.11	0.07	asserted 0.98
s4	In this paper, we prove that the sequence $(\lambda_n)_{n\ge0}$ is strictly infinitely log-concave.	result 0.85	0.97	0.94	asserted 1.00
s5	This resolves a conjecture of Zhu [Math. Z., 2023].	interpretation 0.48	0.69	0.13	asserted 1.00
s6	The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.	method 1.00	0.92	0.03	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (29)

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M. Abramowitz and I. A. Stegun (eds.),Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, DC, 1964

View on Pith 1964
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G. Boros and V. H. Moll,Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004

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P. Brändén,Iterated sequences and the geometry of zeros, J. Reine Angew. Math.658(2011), 115–131

View on Pith 2011
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R. P. Brent, D. J. Platt, and T. S. Trudgian,Accurate estimation of sums over zeros of the Riemann zeta-function, Math. Comp.90(2021), 2923–2935

View on Pith 2021
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R. P. Brent, D. J. Platt, and T. S. Trudgian,The mean square of the error term in the prime number theorem, J. Number Theory238(2022), 740–762

View on Pith 2022
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R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth,On the Lambert Wfunction, Adv. Comput. Math.5(1996), 329–359

View on Pith 1996
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T. C. Craven and G. Csordas,Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math.136(1989), 241–260

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G. Csordas, T. S. Norfolk, and R. S. Varga,The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc.296(1986), 521–541

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G. Csordas, R. S. Varga, and I. Vincze,Jensen polynomials with applications to the Riemann ξ-function, J. Math. Anal. Appl.153(1990), 112–135. 55

View on Pith 1990
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D. K. Dimitrov and F. R. Lucas,Higher order Turán inequalities for the Riemannξ-function, Proc. Amer. Math. Soc.139(2011), 1013–1022

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M. J. Griffin, K. Ono, L. Rolen, J. Thorner, Z. Tripp, and I. Wagner,Jensen polynomials for the Riemann xi-function, Adv. Math.397(2022), Article 108186

View on Pith 2022
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P. R. W. McNamara and B. E. Sagan,Infinite log-concavity: developments and conjectures, Adv. in Appl. Math.44(2010), 1–15

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W. Michałowski,An explicit uniform cubic wedge for consecutive Toeplitz minors of the Rie- mannξ-coefficients, preprint, arXiv:2607.16795 (2026)

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V. H. Moll,The integrals in Gradshteyn and Ryzhik. Part 4: The gamma function, Sci. Ser. A Math. Sci. (N.S.)15(2007), 37–46

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C. O’Sullivan,Zeros of Jensen polynomials and asymptotics for the Riemann xi function, Res. Math. Sci.8(2021), Article 46

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G.Pólya,Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd.7(1927), no. 17, 3–33

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G. Pólya and J. Schur,Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math.144(1914), 89–113

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B.-X. Zhu,On the concavity properties of certain arithmetic sequences and polynomials, Math. Z.305(2023), Article 38. Appendix A. Computer-assisted verification and reproducibility This appendix records the computational details underlying Proposition 4.5, including the soft- ware environment, reproduction command, and implementation used to certify the r...

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

M. Abramowitz and I. A. Stegun (eds.),Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, DC, 1964

View on Pith 1964

[2] [2]

G. Boros and V. H. Moll,Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004

View on Pith 2004

[3] [3]

P. Brändén,Iterated sequences and the geometry of zeros, J. Reine Angew. Math.658(2011), 115–131

View on Pith 2011

[4] [4]

R. P. Brent, D. J. Platt, and T. S. Trudgian,Accurate estimation of sums over zeros of the Riemann zeta-function, Math. Comp.90(2021), 2923–2935

View on Pith 2021

[5] [5]

R. P. Brent, D. J. Platt, and T. S. Trudgian,The mean square of the error term in the prime number theorem, J. Number Theory238(2022), 740–762

View on Pith 2022

[6] [6]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth,On the Lambert Wfunction, Adv. Comput. Math.5(1996), 329–359

View on Pith 1996

[7] [7]

T. C. Craven and G. Csordas,Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math.136(1989), 241–260

View on Pith 1989

[8] [8]

G. Csordas, T. S. Norfolk, and R. S. Varga,The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc.296(1986), 521–541

View on Pith 1986

[9] [9]

G. Csordas, R. S. Varga, and I. Vincze,Jensen polynomials with applications to the Riemann ξ-function, J. Math. Anal. Appl.153(1990), 112–135. 55

View on Pith 1990

[10] [10]

D. K. Dimitrov and F. R. Lucas,Higher order Turán inequalities for the Riemannξ-function, Proc. Amer. Math. Soc.139(2011), 1013–1022

View on Pith 2011

[11] [11]

G. B. Folland,Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989

View on Pith 1989

[12] [12]

M. J. Griffin, K. Ono, L. Rolen, and D. Zagier,Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA116(2019), 11103–11110

View on Pith 2019

[13] [13]

M. J. Griffin, K. Ono, L. Rolen, J. Thorner, Z. Tripp, and I. Wagner,Jensen polynomials for the Riemann xi-function, Adv. Math.397(2022), Article 108186

View on Pith 2022

[14] [14]

F. Haslinger,Complex Analysis: A Functional Analytic Approach, De Gruyter Graduate, De Gruyter, Berlin–Boston, 2018

View on Pith 2018

[15] [15]

P. R. W. McNamara and B. E. Sagan,Infinite log-concavity: developments and conjectures, Adv. in Appl. Math.44(2010), 1–15

View on Pith 2010

[16] [16]

W. Michałowski,An explicit uniform cubic wedge for consecutive Toeplitz minors of the Rie- mannξ-coefficients, preprint, arXiv:2607.16795 (2026)

View on Pith Pith review arXiv 2026

[17] [17]

V. H. Moll,The integrals in Gradshteyn and Ryzhik. Part 4: The gamma function, Sci. Ser. A Math. Sci. (N.S.)15(2007), 37–46

View on Pith 2007

[18] [18]

R. E. Moore, R. B. Kearfott, and M. J. Cloud,Introduction to Interval Analysis, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2009

View on Pith 2009

[19] [19]

C. O’Sullivan,Zeros of Jensen polynomials and asymptotics for the Riemann xi function, Res. Math. Sci.8(2021), Article 46

View on Pith 2021

[20] [20]

G.Pólya,Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd.7(1927), no. 17, 3–33

View on Pith 1927

[21] [21]

G. Pólya and J. Schur,Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math.144(1914), 89–113

View on Pith 1914

[22] [22]

Q. I. Rahman and G. Schmeisser,Analytic Theory of Polynomials, London Mathematical So- ciety Monographs, New Series, vol. 26, Clarendon Press, Oxford University Press, Oxford, 2002

View on Pith 2002

[23] [23]

D. Romik,Orthogonal polynomial expansions for the Riemann xi function in the Hermite, Meixner–Pollaczek, and continuous Hahn bases, Acta Arith.200(2021), 259–329

View on Pith 2021

[24] [24]

W. Rudin,Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976

View on Pith 1976

[25] [25]

W. Rudin,Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987

View on Pith 1987

[26] [26]

M. R. Spiegel, S. Lipschutz, and J. Liu,Schaum’s Mathematical Handbook of Formulas and Tables, McGraw-Hill, New York, 2009

View on Pith 2009

[27] [27]

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