arxiv: 2604.09707 · v1 · submitted 2026-04-07 · 🧮 math.HO

Recognition: 3 theorem links

· Lean Theorem

Analogues of a formula of Ferrar: what I have learned from Semyon Yakubovich

Pedro Ribeiro

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:54 UTC · model grok-4.3

classification 🧮 math.HO

keywords Ferrar's formulasummation formulasMellin transformDirichlet seriesYakubovich

0 comments

The pith

Ferrar's summation formulas connect to Dirichlet series through their functional behavior, enabling new generalizations via the Mellin transform.

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

W. L. Ferrar first linked the functional properties of a summation formula directly to the analytic behavior of its underlying Dirichlet series. Starting from one such formula, the paper describes new generalizations obtained after extensive study with Semyon Yakubovich. It also supplies a concise overview of summation-formula theory, with the Mellin transform acting as the bridge that unifies the author's approach with Yakubovich's contributions. A reader would care because the connection clarifies how summation identities arise from the same transform machinery that governs Dirichlet series, potentially streamlining the discovery of further identities.

Core claim

Ferrar's formula establishes that the functional aspects of a summation formula determine the properties of the associated Dirichlet series, and the Mellin transform serves as the explicit link that permits new analogues to be constructed by varying the underlying series or kernel in a controlled way.

What carries the argument

The Mellin transform, which converts the summation formula into an identity for the Dirichlet series and back, thereby generating analogues by altering the transform pair.

Load-bearing premise

The new generalizations are assumed to follow directly from the same functional connection that Ferrar used, without the paper supplying the explicit verification steps.

What would settle it

An explicit counterexample in which one of the stated generalizations produces a summation formula whose Mellin transform does not recover the expected Dirichlet series identity.

read the original abstract

W. L. Ferrar seems to have been the first mathematician to clearly draw a connection between the functional aspects of a summation formula and the behavior of the Dirichlet series underlying it. Taking a formula due to him as a starting point, I will describe some new generalizations of Ferrar's formulas and how these were actually obtained after learning a great deal from Semyon. I also present a very concise overview of the underlying theory of summation formulas and how the Mellin transform has been the link between mine and Professor Yakubovich's interests.

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

A short personal note describing some generalizations of Ferrar's summation formulas learned from Yakubovich, plus a brief Mellin-transform overview of the theory.

read the letter

This paper is a short personal reflection on some generalizations of W. L. Ferrar's summation formulas. The author explains how these came about after studying with Semyon Yakubovich and provides a concise overview of the theory of summation formulas, emphasizing the role of the Mellin transform in connecting them to Dirichlet series behavior. What the paper does well is to give clear historical context. It points out Ferrar's early recognition of the link between the functional properties of a summation formula and the analytic properties of the associated Dirichlet series. The account of learning from Yakubovich is presented without exaggeration, which makes the narrative credible. The overview section ties together the author's work with Yakubovich's interests in a way that could help readers see the broader picture in this corner of analytic number theory. The soft spots are in the level of detail. The new generalizations are described in outline form, but the paper does not supply the derivations, explicit statements of the formulas, or any verification steps. This is appropriate for a history-of-mathematics piece, yet it means the work does not stand alone as a source for the actual mathematics. Anyone wanting to use or confirm the generalizations would need additional sources or direct communication with the author. There are no computational checks or comparisons with prior results included. This kind of paper suits readers who are already engaged with the history of summation methods or the Mellin transform applications in number theory. It offers context and a personal perspective rather than new technical tools or proofs. A broader audience in mathematics would likely find it too specialized and lacking in substance. The paper shows honest engagement with the literature and avoids overclaiming what it delivers. I think it deserves peer review because the central claims are modest and rest on established connections rather than unverified assertions. A referee could assess whether the generalizations are accurately presented as new and whether the historical points are correct. My recommendation is to send it to peer review. It is a minor but well-intentioned contribution that could appear in a suitable journal as a historical note.

Referee Report

0 major / 0 minor

Summary. The paper describes new generalizations of W. L. Ferrar's summation formulas, obtained after learning from Semyon Yakubovich, building on Ferrar's link between the functional aspects of summation formulas and the behavior of underlying Dirichlet series. It also provides a concise overview of the theory of summation formulas and the connecting role of the Mellin transform between the author's and Yakubovich's interests.

Significance. If the generalizations hold as described, the work offers a useful expository contribution to the history and conceptual development of summation formulas. By documenting the influence of Yakubovich's insights and framing the Mellin transform as a unifying link, it provides context that may assist readers in tracing the evolution of these analytic tools in number theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its content, and the recommendation to accept. The referee's comments correctly identify the paper's focus on new analogues of Ferrar's summation formulas, their derivation following insights from Semyon Yakubovich, and the expository role of the Mellin transform in connecting summation formulas to Dirichlet series.

Circularity Check

0 steps flagged

No significant circularity; expository overview with external attribution

full rationale

The paper is an expository math.HO manuscript that narrates generalizations of Ferrar's formulas learned from Semyon Yakubovich and gives a concise overview of summation formulas connected via the Mellin transform. It presents no formal derivations, equations, fitted parameters, or predictions. The central claims rest on external learning and standard background theory rather than any self-referential reduction, self-citation chain, or ansatz smuggled in. No load-bearing steps exist that could be inspected for circularity by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be identified from the text.

pith-pipeline@v0.9.0 · 5378 in / 987 out tokens · 45479 ms · 2026-05-10T17:54:41.778105+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Ferrar … clearly draw a connection between the functional aspects of a summation formula and the behavior of the Dirichlet series … Mellin transform has been the link
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.1 … rk(n) … Whittaker function W_{1-k/2,0} … functional equation π^{-s}Γ(s)ζ_k(s) = …
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Proof … Mellin-Barnes integral … residue theorem … Phragmén-Lindelöf … Stirling

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

98 extracted references · 98 canonical work pages

[1]

G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part IV, Springer-Verlag, New York, 2013

work page 2013
[2]

B. C. Berndt, Arithmetical identities and Hecke’s functional equation,Proc. Edinburgh Math. Soc.16(1969), 221–226

work page 1969
[3]

B. C. Berndt, Generalized Dirichlet series and Hecke’s functional equation,Proc. Edinburgh Math. Soc.,15(1967), 309–313

work page 1967
[4]

B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. I,Trans. Amer. Math. Soc.,137(1969), 345–359

work page 1969
[5]

B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. II,Trans. Amer. Math. Soc.,137(1969), 361–374

work page 1969
[6]

B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. III.,Trans. Amer. Math. Soc.,146(1969), 323–348. 27

work page 1969
[7]

B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. IV,Trans. Amer. Math. Soc.,149(1970), 179–185

work page 1970
[8]

B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. V.,Trans. Amer. Math. Soc.,160(1971), 139–156

work page 1971
[9]

B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. VI,Trans. Amer. Math. Soc.160(1971), 157-167

work page 1971
[10]

B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. VII,Trans. Amer. Math. Soc.201(1975), 247-261

work page 1975
[11]

B. C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan,Rocky Mountain J. Math.,7(1977), 147–189

work page 1977
[12]

B. C. Berndt, On the Zeros of a Class of Dirichlet Series I,Illinois J. Mathematics,14(1970), 244-258

work page 1970
[13]

B. C. Berndt, On the Zeros of a Class of Dirichlet Series II,Illinois J. Mathematics,14(1970), 678-691

work page 1970
[14]

B. C. Berndt, The number of zeros of the Dedekind zeta-function on the critical line,J. Number Theory3(1971), 1–6

work page 1971
[15]

B. C. Berndt, The number of zeros ofζ(k)(s),J. London Math. Society2(1970), 577-580

work page 1970
[16]

Alladi, J

B.C.Berndt, A.Dixit, AtransformationformulainvolvingtheGammaandRiemannzetafunctionsinRamanujan’sLostNotebook, in: K. Alladi, J. Klauder, C. R. Rao (Eds.), The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Springer, New York, 2010, 199-210

work page 2010
[17]

B. C. Berndt, A. Dixit, R. Gupta, A. Zaharescu, A Class of Identities Associated with Dirichlet Series Satisfying Hecke’s Functional Equation,Proc. Amer. Math. Society,150(2022), 4785-4799

work page 2022
[18]

B. C. Berndt, A. Dixit, R. Gupta, and A. Zaharescu, Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions,Canad. J. Math.(2022), 1—31

work page 2022
[19]

B. C. Berndt, A. Dixit, J. Sohn, Character analogues of theorems of Ramanujan, Koshliakov, and Guinand,Adv. Appl. Math.,46 (2011), 54—70

work page 2011
[20]

B. C. Berndt, R. J. Evans, The determination of Gauss sums,Bull. Amer. Math. Soc.5(1981), 107—129

work page 1981
[21]

Berndt, A

B. Berndt, A. Dixit, R. Gupta, A. Zaharescu, Ramanujan and Koshliakov Meet Abel and Plana, to appear inSome contributions to number theory and beyond: Proceedings of the centenary symposium for M V Subbarao, Fields Institute Communications, Baskar Balasubramanyam, Kaneenika Sinha and Mathukumalli Vidyasagar eds

work page
[22]

B. C. Berndt, A. Dixit, R. Gupta, A. Zaharescu, Modified Bessel functions in Analytic Number Theory, to appear in the volume commemorating Richard Askey, Springer

work page
[23]

B. C. Berndt, A. Dixit, S. Kim and A. Zaharescu, On a theorem of A. I. Popov on sums of squares,Proc. Amer. Math. Soc.,145 (2017), 3795-3808

work page 2017
[24]

B. C. Berndt, A. Dixit, S. Kim, A. Zaharescu, Sums of squares and products of Bessel functions,Advances in Mathematics,338 (2018), 305-338

work page 2018
[25]

B. C. Berndt, S. Kim, A. Zaharescu, The Circle Problem of Gauss and the Divisor Problem of Dirichlet — Still Unsolved,Amer. Math. Monthly125(2018), 99-114

work page 2018
[26]

B. C. Berndt, Y. Lee, and J. Sohn, Koshliakov’s formula and Guinand’s formula in Ramanujan’s lost notebook,Surveys in number theory, Dev. Math.,17(2012), 21-42

work page 2012
[27]

Bochner, Some properties of modular relations,Ann

S. Bochner, Some properties of modular relations,Ann. Math.,53(1951), 332-360

work page 1951
[28]

Bochner, Connection between functional equations and modular relations and functions of exponential type,J

S. Bochner, Connection between functional equations and modular relations and functions of exponential type,J. Indian Math. Soc.16(1952) 99-102

work page 1952
[29]

Bochner, Fourier series came first,Amer

S. Bochner, Fourier series came first,Amer. Math. Monthly86(1979), 197–199

work page 1979
[30]

H. Bohr, E. Landau, Sur les zéros de la fonctionζ(s)de Riemann,Comptes Rendus de l’Acad. des Sciences (Paris)158(1914), 106–110

work page 1914
[31]

Bochner and K

S. Bochner and K. Chandrasekharan, On Riemann’s Functional Equation,Ann. of Math.,63(1956), 336–360

work page 1956
[32]

Yu. A. Brychkov, O. I. Marichev, N. V. Savischenko, Handbook of Mellin Transforms, CRC Press, 2019. 28

work page 2019
[33]

Chandrasekharan and R

K. Chandrasekharan and R. Narasimhan, Hecke’s functional equation and arithmetical identities,Ann. of Math.,74(1961), 1-23

work page 1961
[34]

Chandrasekharan and R

K. Chandrasekharan and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions.Ann. of Math.,76(1962), 93-136

work page 1962
[35]

Davenport, Multiplicative Number Theory, Springer-Verlag, Berlin Heidelberg, 2nd Edition, 1980

H. Davenport, Multiplicative Number Theory, Springer-Verlag, Berlin Heidelberg, 2nd Edition, 1980

work page 1980
[36]

Dixit, Character analogues of Ramanujan type integrals involving the RiemannΞ-function,Pacific J

A. Dixit, Character analogues of Ramanujan type integrals involving the RiemannΞ-function,Pacific J. Math.,255(2012), 317-348

work page 2012
[37]

Dixit and R

A. Dixit and R. Gupta, Koshliakov zeta functions I. Modular relations,Adv. Math.393(2021), Paper No. 108093

work page 2021
[38]

Dixit, Analogues of the general theta transformation formula,Proc

A. Dixit, Analogues of the general theta transformation formula,Proc. Roy. Soc. Edinburgh, Sect. A,143(2013), 371–399

work page 2013
[39]

Dixit, Ramanujan’s ingenious method for generating modular-type transformation formulas, in: The Legacy of Srinivasa Ramanujan, in: RMS-Lecture Note Series,20(2013) pp

A. Dixit, Ramanujan’s ingenious method for generating modular-type transformation formulas, in: The Legacy of Srinivasa Ramanujan, in: RMS-Lecture Note Series,20(2013) pp. 163–179

work page 2013
[40]

Dixit, Series transformations and integrals involving the RiemannΞ-function,J

A. Dixit, Series transformations and integrals involving the RiemannΞ-function,J. Math. Anal. Appl.368(2010), 358–373

work page 2010
[41]

A. L. Dixon, W. L. Ferrar, Lattice-point summation formulae,Quart. J. Math. Oxford,2(1931), 31-54

work page 1931
[42]

A. L. Dixon, W. L. Ferrar, Some summations over the lattice points of a circle I.,Quart. J. Math. Oxford,5(1934), 48-63

work page 1934
[43]

A. L. Dixon, W. L. Ferrar, Some summations over the lattice points of a circle II.,Quart. J. Math. Oxford,5(1934), 172-185

work page 1934
[44]

A. L. Dixon and W. L. Ferrar, On the summation formulae of Voronoï and Poisson,Quart. J. Math. Oxford8(1937), 66-74

work page 1937
[45]

H. M. Edwards, Riemann’s Zeta Function, 1974, New York: Academic Press

work page 1974
[46]

Epstein, Zur Theorie allgemeiner Zetafunctionen,Math

P. Epstein, Zur Theorie allgemeiner Zetafunctionen,Math. Ann.56(1903), 615-644

work page 1903
[47]

Erdélyi, W

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vols.1,2and3, McGraw-Hill, New York, London and Toronto, 1953

work page 1953
[48]

W. L. Ferrar, Some solutions of the equationF(t) =F(t−1),J. London Math. Soc.,11(1936), 99–103

work page 1936
[49]

W. L. Ferrar, Summation formulae and their relation to Dirichlet’s series I.,Compositio Math.,1(1935), 344-360

work page 1935
[50]

W. L. Ferrar, Summation formulae and their relation to Dirichlet’s series II.,Compositio Math.,4(1937), 394-405

work page 1937
[51]

A. P. Guinand, On Poisson’s summation formula,Ann. Math.,42(1941), 591-603

work page 1941
[52]

A. P. Guinand, Summation formulae and self-reciprocal functions. I,Quart. J. Math

work page
[53]

A. P. Guinand, Summation formulae and self-reciprocal functions (II),Quart. J. Math.,1(1939), 104—118

work page 1939
[54]

A. P. Guinand, Some rapidly convergent series for the Riemannζ-function,Quart J. Math.(Oxford)6(1955), 156—160

work page 1955
[55]

Emil Grosswald, Comments on some formulae of Ramanujan,Acta Arith.,21(1972), 25-34

work page 1972
[56]

G. H. Hardy, Sur les zéros de la fonctionζ(s)de Riemann,Comptes rendus,158(1914), 1012-1014

work page 1914
[57]

Kanemitsu, Y

S. Kanemitsu, Y. Tanigawa, and M. Yoshimoto, Ramanujan’s formula and modular relations, in Number theoretic methods: future trends (S. Kanemitsu and C. Jia, eds.), Kluwer, Dordrecht, 2002, 159-212

work page 2002
[58]

N. S. Koshliakov, (under the name N. S. Sergeev), Issledovanie odnogo klassa transtsendentnykh funktsii, opredelyaemykh obob- shcennym yravneniem Rimana (A study of a class of transcendental functions defined by the generalized Riemann equation) (in Russian), Trudy Mat. Inst. Steklov, Moscow, 1949, available online at https://dds .crl .edu /crldelivery /14052

work page 1949
[59]

Landau, Über die Hardysche Entdeckung unendlich vieler Nullstellen der Zetafunktion mit reellem Teil1 2,Math

E. Landau, Über die Hardysche Entdeckung unendlich vieler Nullstellen der Zetafunktion mit reellem Teil1 2,Math. Annalen,76 (1915), 212-243

work page 1915
[60]

Levinson, Almost all roots ofζ(s) =aare arbitrarily close toσ= 1/2,Proc

N. Levinson, Almost all roots ofζ(s) =aare arbitrarily close toσ= 1/2,Proc. Nat. Acad. Sci. U.S.A.72(1975), 1322–1324

work page 1975
[61]

Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983

O.I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983

work page 1983
[62]

L. J. Mordell, Some applications of Fourier series in the analytic theory of numbers,Proc. Cambridge Philos. Soc.(1928), 585-596

work page 1928
[63]

L. J. Mordell, Poisson’s summation formula and the Riemann zeta-function,J. London Math. Soc.4(1929), 285-291. 29

work page 1929
[64]

Nasim, A summation formula involvingσk(n), k >1,Canad

C. Nasim, A summation formula involvingσk(n), k >1,Canad. J. Math.21(1969), 951-964

work page 1969
[65]

Nasim, A summation formula involvingσ(n),Trans

C. Nasim, A summation formula involvingσ(n),Trans. Amer. Math. Soc.192(1974), 307—317

work page 1974
[66]

Nasim, On the summation formula of Voronoï,Trans

C. Nasim, On the summation formula of Voronoï,Trans. Amer. Math. Soc.,163(1972), 35-45

work page 1972
[67]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010

work page 2010
[68]

Oberhettinger and K

F. Oberhettinger and K. L. Soni, On some relations which are equivalent to functional equations involving the Riemann zeta- function,Math. Z.127(1972), 17–34

work page 1972
[69]

T. L. Pearson, Note on the Hardy-Landau summation formula,Canad. Math. Bull.,8(1965), 717-720

work page 1965
[70]

A. I. Popov, On some summation formulas (in Russian), Bull. Acad. Sci. URSS,6(1934), 801–802 (in Russian). This paper can be found here

work page 1934
[71]

A. I. Popov, Über die zylindrische Funktionen enthaltenden Reihen,C. R. Acad. Sci. URSS20(1935), 96–99 (in Russian)

work page 1935
[72]

H. S. A. Potter, E. C. Titchmarsh, The zeros of Epstein’s zeta-functions,Proc. London Math. Soc. (2),39(1935), 372-384

work page 1935
[73]

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series: Vol.1: Elementary Functions, Gordon and Breach, New York, 1986; Vol.2: Special Functions, Gordon and Breach, New York, 1986; Vol.3: More Special Functions, Gordon and Breach, New York, 1989

work page 1986
[74]

Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988

work page 1988
[75]

Ribeiro, Another Classic Fourier Analysis Proof of the Basel Problem,The Mathematical Gazette103(2019), 142-145

P. Ribeiro, Another Classic Fourier Analysis Proof of the Basel Problem,The Mathematical Gazette103(2019), 142-145

work page 2019
[76]

Ribeiro, Another proof of the Famous Formula for the Zeta Function at the Positive Even integersAmer

P. Ribeiro, Another proof of the Famous Formula for the Zeta Function at the Positive Even integersAmer. Math. Monthly125 (2018), 839-841

work page 2018
[77]

Ribeiro, Summation and Transformation Formulas related with Special Functions, MSc Thesis, Faculdade de Ciências da Universidade do Porto, Portugal

P. Ribeiro, Summation and Transformation Formulas related with Special Functions, MSc Thesis, Faculdade de Ciências da Universidade do Porto, Portugal

work page
[78]

Ribeiro, S

P. Ribeiro, S. Yakubovich, Certain extensions of results of Siegel, Wilton and Hardy,Adv. Appl. Math.,156(2024), 102676, 94 pages

work page 2024
[79]

Ribeiro, S

P. Ribeiro, S. Yakubovich, Generalizations (in the spirit of Koshliakov) of some formulas from Ramanujan’s lost notebook. In: Castillo, K., Durán, A.J. (eds) Orthogonal Polynomials and Special Functions. Coimbra Mathematical Texts, vol 3. Springer, Cham

work page
[80]

Ribeiro, S Yakubovich, On the Epstein zeta function and the zeros of a class of Dirichlet series, to appear in theJournal of Math

P. Ribeiro, S Yakubovich, On the Epstein zeta function and the zeros of a class of Dirichlet series, to appear in theJournal of Math. Analysis and Applications,530(2024), 127590, 44 pages

work page 2024

Showing first 80 references.