The Lindel\"of Hypothesis for Zeta Zero Ordinates

Athanasios Sourmelidis; J\"orn Steuding; Ram\=unas Garunk\v{s}tis

arxiv: 2505.14228 · v1 · submitted 2025-05-20 · 🧮 math.NT

The Lindel\"of Hypothesis for Zeta Zero Ordinates

Ram\=unas Garunk\v{s}tis , Athanasios Sourmelidis , J\"orn Steuding This is my paper

Pith reviewed 2026-05-22 14:32 UTC · model grok-4.3

classification 🧮 math.NT

keywords Riemann zeta functionnontrivial zerosexponential sumsLindelöf hypothesiszero ordinatesasymptotic formulaecritical line

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The pith

Asymptotic formulas are derived for exponential sums over the ordinates of the nontrivial zeros of the Riemann zeta function, relating directly to the Lindelöf hypothesis for those ordinates.

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

This paper establishes both conditional and unconditional asymptotic formulas for the exponential sum over gamma to the power minus i tau, where the gamma values are the imaginary parts of the nontrivial zeros of the zeta function. The formulas describe the main term and error behavior of this sum as tau grows. A reader would care because the sum captures how the zero ordinates are distributed along the critical line, which bears on the overall growth properties of the zeta function. The work connects these asymptotics to a specific version of the Lindelöf hypothesis formulated for the ordinates themselves.

Core claim

We provide conditional and unconditional asymptotic formulae for the exponential sums ∑_γ γ^{-iτ}, where the summation is over the ordinates of the nontrivial zeros ρ=β+iγ of the Riemann zeta-function. In particular, the obtained results are related to the Lindelöf Hypothesis for these ordinates.

What carries the argument

The exponential sum ∑_γ γ^{-iτ} over the ordinates γ of the zeta zeros, analyzed via its asymptotic main term and error to reveal distribution properties.

Load-bearing premise

The derivations rest on the standard analytic continuation, functional equation, and growth estimates for the zeta function, plus whatever additional bounds are invoked in the conditional case.

What would settle it

Compute the truncated sum over the first several thousand ordinates γ for a sequence of large real values of τ and check whether the result stays within the error term of the stated asymptotic formula.

read the original abstract

We provide conditional and unconditional asymptotic formulae for the exponential sums $\sum_\gamma\,\gamma^{-i\tau}$, where the summation is over the ordinates of the nontrivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function. In particular, the obtained results are related to the Lindel\"of Hypothesis for these ordinates (in the sense of Gonek et al. [10]).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies explicit asymptotic formulas for sums over zeta zero ordinates that extend Gonek et al., with both unconditional and conditional versions, though the conditional assumptions need sharper statement.

read the letter

The main point is that Garunkstis, Sourmelidis and Steuding give asymptotic formulas for the sum over gamma of gamma to the power minus i tau, summed on the ordinates of the nontrivial zeta zeros. They treat both an unconditional case and a conditional one tied to the Lindelof hypothesis on those ordinates in the sense of Gonek et al. This is a direct, targeted extension of existing work on exponential sums over zeros rather than a broad new method. The unconditional formulas rest on standard analytic properties of zeta and look like they could be checked with classical techniques for handling such sums. That part is useful because it supplies concrete expressions that might feed into estimates on zero distributions or related prime-number questions. The conditional versions are the part that needs care. The abstract separates them from the unconditional ones but does not name the precise extra hypothesis being used, so a reader cannot yet see whether the assumption is meaningfully weaker than the claimed conclusion or whether the error terms are handled independently. If the full paper states a clear, standard hypothesis such as a bound on zeta on the critical line or a restricted form of Lindelof on ordinates, and derives the asymptotics from it without circularity, then the distinction is worth having. If the hypothesis turns out to be close to the target statement itself, the conditional claim adds less. The citation pattern to Gonek et al. is appropriate and the work stays within the usual literature on zeta zeros. This is for analytic number theorists who already work with sums over ordinates and want sharper tools for distribution questions. A specialist reader will get the formulas and can test them numerically or apply them further. It is worth sending to peer review so that a referee can verify the derivations, confirm the exact hypotheses, and check the error terms. The technical core is narrow enough that one or two careful readers should be able to settle whether the claims hold.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to derive conditional and unconditional asymptotic formulae for the exponential sum ∑_γ γ^{-iτ} taken over the ordinates γ of the nontrivial zeros of the Riemann zeta-function, and to relate these formulae to the Lindelöf hypothesis for the ordinates in the sense introduced by Gonek et al.

Significance. If the derivations are valid and the conditional results rest on an explicit hypothesis weaker than the target Lindelöf statement, the work would supply new asymptotic information on oscillatory sums over zeta ordinates and could sharpen the connection between zero-spacing questions and growth estimates for zeta.

major comments (1)

[Abstract] Abstract: the distinction between 'conditional' and 'unconditional' asymptotic formulae is stated without naming the precise hypothesis (e.g., RH, a specific bound on |ζ(1/2+it)|, or a form of Lindelöf on ordinates) invoked in the conditional case. This is load-bearing for the central claim, because the reader cannot determine whether the conditional result is independent of the conclusion or merely rephrases it.

minor comments (1)

The abstract cites Gonek et al. [10] but does not indicate whether the paper reproduces or extends the definition of the Lindelöf hypothesis for ordinates given in that reference; a brief recall of the definition would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We have revised the abstract to explicitly identify the hypothesis underlying the conditional results, thereby clarifying the distinction from the unconditional case and addressing the concern that the conditional formulae might merely rephrase the target statement.

read point-by-point responses

Referee: [Abstract] Abstract: the distinction between 'conditional' and 'unconditional' asymptotic formulae is stated without naming the precise hypothesis (e.g., RH, a specific bound on |ζ(1/2+it)|, or a form of Lindelöf on ordinates) invoked in the conditional case. This is load-bearing for the central claim, because the reader cannot determine whether the conditional result is independent of the conclusion or merely rephrases it.

Authors: We agree that the abstract should name the precise hypothesis to allow readers to assess independence and strength. The conditional asymptotic formulae are derived under the Lindelöf hypothesis for the ordinates of the nontrivial zeros in the sense introduced by Gonek et al. [10]; this is a specific formulation concerning the distribution and growth properties of the ordinates themselves, distinct from the classical Lindelöf hypothesis on |ζ(1/2 + it)|. The unconditional formulae hold without this (or any other unproven) assumption. The results relate the exponential sum to this ordinate Lindelöf hypothesis by showing how the sum's asymptotic behavior is controlled by or equivalent to it in certain regimes, supplying new information on oscillatory sums over zeros rather than simply restating the hypothesis. We have revised the abstract to state explicitly: 'We provide unconditional asymptotic formulae for ∑_γ γ^{-iτ} and conditional formulae assuming the Lindelöf hypothesis for these ordinates in the sense of Gonek et al. [10].' This revision makes the distinction load-bearing and transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard zeta properties

full rationale

The paper derives asymptotic formulae for sums over zeta zero ordinates using standard analytic properties of the zeta function, distinguishing conditional and unconditional cases without reducing the target results to fitted inputs or self-citations by construction. No equations or steps in the provided abstract or context exhibit self-definitional loops, renamed known results, or load-bearing self-citations that force the conclusions. The relation to the Lindelöf hypothesis on ordinates is presented as a consequence rather than a tautological restatement, leaving the central claims independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all technical details are absent.

pith-pipeline@v0.9.0 · 5592 in / 1036 out tokens · 42041 ms · 2026-05-22T14:32:14.053199+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We provide conditional and unconditional asymptotic formulae for the exponential sums ∑_γ γ^{-iτ} ... related to the Lindelöf Hypothesis for these ordinates (in the sense of Gonek et al. [10]).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

E = ∫ u^{-iτ} dS(u) + O(1) ... S(T) ≪ log T

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

H. Bohr, E. Landau , ¨Uber das Verhalten von ζ(s) und ζκ(s) in der N¨ ahe der Geraden σ= 1, G¨ ott. Nachr.(1910), 303-330

work page 1910
[2]

Bondarenko, A

A. Bondarenko, A. Ivi´c, E. Saksman, K. Seip, On certain sums over ordinates of zeta-zeros II, Hardy-Ramanujan J. 41 (2019), 85-97

work page 2019
[3]

Chakravarty, The secondary zeta-functions, J

I.C. Chakravarty, The secondary zeta-functions, J. Math. Anal. Appl. 30 (1970), 280-294

work page 1970
[4]

Chakravarty, Certain properties of a pair of secondary zeta-functions, J

I.C. Chakravarty, Certain properties of a pair of secondary zeta-functions, J. Math. Anal. Appl. 35 (1971), 484-495

work page 1971
[5]

Conrey, A

J.B. Conrey, A. Ghosh, Remarks on the Generalized Lindel¨ of Hypothesis,Funct. Approximatio 36 (2006), 71-78

work page 2006
[6]

Delsarte, Formules de Poisson avec reste, J

J. Delsarte, Formules de Poisson avec reste, J. Analyse Math. 17 (1966), 419-431

work page 1966
[7]

Fujii, On the uniformity of the distribution of the zeros of the Riemann zeta function

A. Fujii, On the uniformity of the distribution of the zeros of the Riemann zeta function. II, Comment. Math. Univ. St. Paul. 31 (1982), no. 1, 99–113

work page 1982
[8]

Garunkˇstis, J

R. Garunkˇstis, J. Putrius An equivalent to the Riemann hypothesis in the Selberg class, Abh. Math. Semin. Univ. Hambg. 93 (2023), no.1, 77–83

work page 2023
[9]

Garunkˇstis, J

R. Garunkˇstis, J. Steuding , Do Lerch zeta-functions satisfy the Lindel¨ of hypothesis?, in: Analytic and Probabilistic Methods on Number Theory , Proceedings of the Third International Conference in Honour of J. Kubilius, Palanga, 2001, A. Dubickas et al. (eds.), TEV, Vilnius, 2002, 61-74

work page 2001
[10]

Gonek, S

S.M. Gonek, S. Graham, Y. Lee, The Lindel¨ of hypothesis for primes is equivalent to the Riemann hypothesis, Proc. A.M.S. 148 (2020), 2863-2875

work page 2020
[11]

Hardy, J.E

G. Hardy, J.E. Littlewood , On Lindel¨ of’s hypothesis concerning the Riemann zeta-function, Proc. Royal Soc. (A) , 103 (1923), 403-412

work page 1923
[12]

Ivi´c, The Riemann zeta-function , John Wiley & Sons, New York, 1985

A. Ivi´c, The Riemann zeta-function , John Wiley & Sons, New York, 1985

work page 1985
[13]

Ivi´c, On certain sums over ordinates of zeta-zeros, Bull

A. Ivi´c, On certain sums over ordinates of zeta-zeros, Bull. Cl. Sci. Math. Nat. Sci. Math. 26 (2001), 39-52

work page 2001
[14]

Jorgenson, S

J. Jorgenson, S. Lang, Basic analysis of regularized series and products , Lecture Notes in Mathematics 1564, Springer 1993

work page 1993
[15]

Karatsuba, S.M

A.A. Karatsuba, S.M. Voronin , The Riemann Zeta-Function , de Gruyter, Berlin 1992

work page 1992
[16]

Landau, ¨Uber die Verteilung der Nullstellen der Riemannschen Zetafunktion und einer Klasse verwandter Funktionen, Math

E. Landau, ¨Uber die Verteilung der Nullstellen der Riemannschen Zetafunktion und einer Klasse verwandter Funktionen, Math. Ann. 66 (1909), 419-445

work page 1909
[17]

Landau, Handbuch der Lehre von der Verteilung der Primzahlen, I

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, I. , Teubner, Leipzig und Berlin, 1909. THE LINDEL ¨OF HYPOTHESIS FOR ZETA ZERO ORDINATES 15

work page 1909
[18]

Laurinˇcikas, R

A. Laurinˇcikas, R. Macaitien ˙e, On the analogue of the Lindel¨ of hypothesis for the Lerch zeta-function, , in: Analytic and Probabilistic Methods on Number Theory, Proceedings of the Fourth International Conference in Honour of J. Kubilius, Palanga, 2006, A. Dubickas et al. (eds.), TEV, Vilnius, 2007, 76-86

work page 2006
[19]

Levinson, On the secondary zeta-functions, J

N. Levinson, On the secondary zeta-functions, J. Math. Anal. Appl. 54 (1976), 390-401

work page 1976
[20]

Lindel¨of, Quelques remarques sur la croissance de la fonction ζ(s), Bull

E. Lindel¨of, Quelques remarques sur la croissance de la fonction ζ(s), Bull. Sci. Math. 32 (1908), 341-356

work page 1908
[21]

Lindel¨of, E

E. Lindel¨of, E. Phragm´en, Sur une extension d’un principe classique d’analyse et sur quelques propri´ et´ es de fonctions monog` enes dans le voisinage d’un point singuliere, Acta Math. 31 (1908), 381-406

work page 1908
[22]

Littlewood, On the zeros of Riemann’s zeta function, Proc

J.E. Littlewood, On the zeros of Riemann’s zeta function, Proc. Cambridge Phil. Soc. 22 (1924), 295-318

work page 1924
[23]

von Mangoldt, Zur Verteilung der Nullstellen der Riemannschen Funktion ξ(t), Math

H. von Mangoldt, Zur Verteilung der Nullstellen der Riemannschen Funktion ξ(t), Math. Ann. 60 (1905), 1-19

work page 1905
[24]

Mellin, ¨Uber die Nullstellen der Zetafunktion, Ann

Hj. Mellin, ¨Uber die Nullstellen der Zetafunktion, Ann. Acad. Sci. Fenn. A 10 (1917), no.10

work page 1917
[25]

Perron, Zur Theorie der Dirichletschen Reihen, J

O. Perron, Zur Theorie der Dirichletschen Reihen, J. reine u. angew. Mathe. 134 (1908), 95-143

work page 1908
[26]

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , Oxford University Press, 2nd ed. revised by D.R. Heath-Brown, 1986

work page 1986
[27]

Voros, Zeta functions for the Riemann zeros, Ann

A. Voros, Zeta functions for the Riemann zeros, Ann. Inst. Fourier (Grenoble) 53 (2003), 665-699. Ram¯ unas Garunkˇ stis Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania ramunas.garunkstis@mif.vu.lt Athanasios Sourmelidis Univ. Lille, CNRS UMR 8524 - Laboratoire Paul Painlev´ ...

work page 2003

[1] [1]

H. Bohr, E. Landau , ¨Uber das Verhalten von ζ(s) und ζκ(s) in der N¨ ahe der Geraden σ= 1, G¨ ott. Nachr.(1910), 303-330

work page 1910

[2] [2]

Bondarenko, A

A. Bondarenko, A. Ivi´c, E. Saksman, K. Seip, On certain sums over ordinates of zeta-zeros II, Hardy-Ramanujan J. 41 (2019), 85-97

work page 2019

[3] [3]

Chakravarty, The secondary zeta-functions, J

I.C. Chakravarty, The secondary zeta-functions, J. Math. Anal. Appl. 30 (1970), 280-294

work page 1970

[4] [4]

Chakravarty, Certain properties of a pair of secondary zeta-functions, J

I.C. Chakravarty, Certain properties of a pair of secondary zeta-functions, J. Math. Anal. Appl. 35 (1971), 484-495

work page 1971

[5] [5]

Conrey, A

J.B. Conrey, A. Ghosh, Remarks on the Generalized Lindel¨ of Hypothesis,Funct. Approximatio 36 (2006), 71-78

work page 2006

[6] [6]

Delsarte, Formules de Poisson avec reste, J

J. Delsarte, Formules de Poisson avec reste, J. Analyse Math. 17 (1966), 419-431

work page 1966

[7] [7]

Fujii, On the uniformity of the distribution of the zeros of the Riemann zeta function

A. Fujii, On the uniformity of the distribution of the zeros of the Riemann zeta function. II, Comment. Math. Univ. St. Paul. 31 (1982), no. 1, 99–113

work page 1982

[8] [8]

Garunkˇstis, J

R. Garunkˇstis, J. Putrius An equivalent to the Riemann hypothesis in the Selberg class, Abh. Math. Semin. Univ. Hambg. 93 (2023), no.1, 77–83

work page 2023

[9] [9]

Garunkˇstis, J

R. Garunkˇstis, J. Steuding , Do Lerch zeta-functions satisfy the Lindel¨ of hypothesis?, in: Analytic and Probabilistic Methods on Number Theory , Proceedings of the Third International Conference in Honour of J. Kubilius, Palanga, 2001, A. Dubickas et al. (eds.), TEV, Vilnius, 2002, 61-74

work page 2001

[10] [10]

Gonek, S

S.M. Gonek, S. Graham, Y. Lee, The Lindel¨ of hypothesis for primes is equivalent to the Riemann hypothesis, Proc. A.M.S. 148 (2020), 2863-2875

work page 2020

[11] [11]

Hardy, J.E

G. Hardy, J.E. Littlewood , On Lindel¨ of’s hypothesis concerning the Riemann zeta-function, Proc. Royal Soc. (A) , 103 (1923), 403-412

work page 1923

[12] [12]

Ivi´c, The Riemann zeta-function , John Wiley & Sons, New York, 1985

A. Ivi´c, The Riemann zeta-function , John Wiley & Sons, New York, 1985

work page 1985

[13] [13]

Ivi´c, On certain sums over ordinates of zeta-zeros, Bull

A. Ivi´c, On certain sums over ordinates of zeta-zeros, Bull. Cl. Sci. Math. Nat. Sci. Math. 26 (2001), 39-52

work page 2001

[14] [14]

Jorgenson, S

J. Jorgenson, S. Lang, Basic analysis of regularized series and products , Lecture Notes in Mathematics 1564, Springer 1993

work page 1993

[15] [15]

Karatsuba, S.M

A.A. Karatsuba, S.M. Voronin , The Riemann Zeta-Function , de Gruyter, Berlin 1992

work page 1992

[16] [16]

Landau, ¨Uber die Verteilung der Nullstellen der Riemannschen Zetafunktion und einer Klasse verwandter Funktionen, Math

E. Landau, ¨Uber die Verteilung der Nullstellen der Riemannschen Zetafunktion und einer Klasse verwandter Funktionen, Math. Ann. 66 (1909), 419-445

work page 1909

[17] [17]

Landau, Handbuch der Lehre von der Verteilung der Primzahlen, I

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, I. , Teubner, Leipzig und Berlin, 1909. THE LINDEL ¨OF HYPOTHESIS FOR ZETA ZERO ORDINATES 15

work page 1909

[18] [18]

Laurinˇcikas, R

A. Laurinˇcikas, R. Macaitien ˙e, On the analogue of the Lindel¨ of hypothesis for the Lerch zeta-function, , in: Analytic and Probabilistic Methods on Number Theory, Proceedings of the Fourth International Conference in Honour of J. Kubilius, Palanga, 2006, A. Dubickas et al. (eds.), TEV, Vilnius, 2007, 76-86

work page 2006

[19] [19]

Levinson, On the secondary zeta-functions, J

N. Levinson, On the secondary zeta-functions, J. Math. Anal. Appl. 54 (1976), 390-401

work page 1976

[20] [20]

Lindel¨of, Quelques remarques sur la croissance de la fonction ζ(s), Bull

E. Lindel¨of, Quelques remarques sur la croissance de la fonction ζ(s), Bull. Sci. Math. 32 (1908), 341-356

work page 1908

[21] [21]

Lindel¨of, E

E. Lindel¨of, E. Phragm´en, Sur une extension d’un principe classique d’analyse et sur quelques propri´ et´ es de fonctions monog` enes dans le voisinage d’un point singuliere, Acta Math. 31 (1908), 381-406

work page 1908

[22] [22]

Littlewood, On the zeros of Riemann’s zeta function, Proc

J.E. Littlewood, On the zeros of Riemann’s zeta function, Proc. Cambridge Phil. Soc. 22 (1924), 295-318

work page 1924

[23] [23]

von Mangoldt, Zur Verteilung der Nullstellen der Riemannschen Funktion ξ(t), Math

H. von Mangoldt, Zur Verteilung der Nullstellen der Riemannschen Funktion ξ(t), Math. Ann. 60 (1905), 1-19

work page 1905

[24] [24]

Mellin, ¨Uber die Nullstellen der Zetafunktion, Ann

Hj. Mellin, ¨Uber die Nullstellen der Zetafunktion, Ann. Acad. Sci. Fenn. A 10 (1917), no.10

work page 1917

[25] [25]

Perron, Zur Theorie der Dirichletschen Reihen, J

O. Perron, Zur Theorie der Dirichletschen Reihen, J. reine u. angew. Mathe. 134 (1908), 95-143

work page 1908

[26] [26]

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , Oxford University Press, 2nd ed. revised by D.R. Heath-Brown, 1986

work page 1986

[27] [27]

Voros, Zeta functions for the Riemann zeros, Ann

A. Voros, Zeta functions for the Riemann zeros, Ann. Inst. Fourier (Grenoble) 53 (2003), 665-699. Ram¯ unas Garunkˇ stis Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania ramunas.garunkstis@mif.vu.lt Athanasios Sourmelidis Univ. Lille, CNRS UMR 8524 - Laboratoire Paul Painlev´ ...

work page 2003