A Product Identity For Dirichlet Series Satisfying Hecke's Functional Equation

Efe G\"urel

REVIEW 2 major objections 4 minor 15 references

A Product Identity For Dirichlet Series Satisfying Hecke's Functional Equation

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For Hecke-series Dirichlet functions, a product of two values equals residue terms plus Bessel-integral sums.

desk verdict The central product identity is false as stated: the Bessel integrals diverge on the claimed domain and Equation (10) silently drops a nonzero constant. read the letter →

arxiv 2504.14551 v1 pith:GMFQF7ML submitted 2025-04-20 math.NT

Efe Gürel This is my paper · ORCID

classification math.NT MSC 11M4111M06

keywords Wilton'sformulaHeckeseriesfunctionalequationDirichletMeijerG-functionBesselintegralsL-functionsRiemannzetafunction

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

Wilton's product formula expresses a product of two Riemann zeta values as a residue term plus an infinite sum of tail integrals. This paper claims the same shape for every Dirichlet series satisfying Hecke's functional equation: Theorem 1.3 writes $\varphi(u)\psi(v)$ as two residue terms plus two absolutely convergent sums of finite-interval Bessel integrals. The same identity governs Hecke series, L-functions of modular forms, Ramanujan's tau L-function, Epstein zeta functions, Dedekind zeta functions of imaginary quadratic fields, and Dirichlet L-functions. The point of the result is that one uniform decomposition replaces case-by-case approximate functional equations and, for the Riemann zeta function, specialises to a four-term product identity.

What carries the argument

The engine is a two-sided Perron-like operator $F_a(\varphi(u),\psi(v);x)$, a vertical-line integral of $\varphi(u+w)\psi(v-w)x^{w+1}/w(w+1)$, together with the Hecke functional equation $\varphi(s)=\gamma(2\pi/\lambda)^{2s-k}\Gamma(k-s)/\Gamma(s)\psi(k-s)$. After a contour shift, the kernel becomes a ratio of gamma functions, which is a Meijer G-function (a Mellin-Barnes integral whose integrand is a product of gamma factors); differentiating that kernel in $x$ collapses it to a Bessel function $J_{k-1}(4\pi\sqrt{nt}/\lambda)$. This is what turns the abstract Dirichlet-series product into the finite-interval Bessel sums in the final identity. The residue terms in the formula come from the poles crossed at $w=0$, $w=k-u$, and $w=-1$ during the contour shift.

What would settle it

Take the Ramanujan tau L-function ($k=12$, cusp form) and set $u=v=14$. The stated identity's Bessel integrand behaves like $t^{12-1-14}=t^{-3}$ as $t\to 0$, so each $\int_0^1$ term diverges. Since $\varphi(14)^2$ is finite, the theorem's formula cannot hold literally unless a regularisation of the divergent integrals is supplied; finding or refuting that regularisation settles the claim.

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Extended reading notes

Core claim

The central discovery is Theorem 1.3, a Wilton-type product formula for two Dirichlet series $\varphi$ and $\psi$ related by Hecke's functional equation of signature $(\lambda,k,\gamma)$. For $\mathrm{Re}(u),\mathrm{Re}(v)>\max(c+1,k)$ and $u,v\neq k+1$, it claims $$\varphi(u)\psi(v)=\frac{\mathrm{Res}\,\varphi(k)}{u-k}\psi(u+v-k)+\frac{\mathrm{Res}\,\psi(k)}{v-k}\varphi(u+v-k)-\frac{2\pi\gamma}{\$\lambda$}\sum_{n=1}^{\infty}\sigma_{\$\beta$,k-u-v}(n)$n^{{1-k/2}}$\$int_0^{1}$ $t^{{k/2-1/2-u}}$J_{k-1}(4\pi\sqrt{nt}/\$\lambda$)\,dt-\frac{2\pi}{\$\lambda$\gamma}\sum_{n=1}^{\infty}\sigma_{\$\alpha$,k-u-v}(n)$n^{{1-k/2}}$\$int_0^{1}$ $t^{{k/2-1/2-v}}$J_{k-1}(4\pi\sqrt{nt}/\$\lambda$)\,dt.$$ The residues are explicit in terms of the constant coefficients and a gamma factor. The proof shifts the contour in a Perron-like integral, applies the Hecke functional equation, expands the product of two $\psi$'s into the convolution $\sigma_{\beta,k-u-v}(n)$, evaluates the resulting Mellin-Barnes integral as a Meijer G-function, and differentiates to land on Bessel functions of order $k-1$. The paper then specialises the formula to modular-form L-functions, Ramanujan's tau function, Epstein zeta functions, Dedekind zeta functions of imaginary quadratic fields, Dirichlet L-functions, and the Riemann zeta function.

Load-bearing premise

The proof's load-bearing step is integrating the Bessel derivative in equation (10) from 0 to x to recover the Meijer G-function; that integration needs $\mathrm{Re}(u)<k$ and $\mathrm{Re}(v)<k$, while the theorem's stated domain $\mathrm{Re}(u),\mathrm{Re}(v)>\max(c+1,k)$ has both variables in the range where the Bessel integrals diverge.

Editorial extensions

If this is right

For cuspidal modular-form L-functions the residue terms vanish, so the product and the square of the L-function are expressed purely as absolutely convergent Bessel sums; the Ramanujan tau L-function is the worked example.
For the normalized Eisenstein series the product formula rearranges to a four-term identity for $\zeta(u)\zeta(v)\zeta(u-k+1)\zeta(v-k+1)$, the advertised Riemann-zeta product identity.
For even and odd primitive Dirichlet characters the Bessel functions of orders $-1/2$ and $1/2$ reduce to cosine and sine, converting the general identity into explicit trigonometric-integral formulas for $L(u,\chi)L(v,\bar{\chi})$.
For Epstein zeta functions of a positive definite quadratic form $Q$ and its inverse, the identity links $Z(u;Q)Z(v;Q^{-1})$ to the two individual zeta functions plus Bessel sums.
Specialising to $u=v$ gives a uniform square formula $\varphi(u)^2=\frac{2}{u-k}\mathrm{Res}\,\varphi(k)\varphi(2u-k)-\frac{4\pi\gamma}{\lambda}\sum_n \sigma_{\alpha,k-2u}(n)n^{1-k/2}\int_0^1 t^{k/2-1/2-u}J_{k-1}(4\pi\sqrt{nt}/\lambda)\,dt$.

Reading between the lines

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

A reader checking equation (10) will find that the integration from 0 to x is valid only when $\mathrm{Re}(u)<k$ and $\mathrm{Re}(v)<k$, whereas the theorem is stated for $\mathrm{Re}(u),\mathrm{Re}(v)>\max(c+1,k)$; the formula could only survive in that range through an unstated regularisation or analytic continuation of the Bessel sums.
Because the proof uses only the Hecke functional equation, the gamma ratio, and the convolution structure of the coefficients, the same two-residue-plus-Bessel shape is likely to hold for any Selberg-class Dirichlet series whose functional equation is of Hecke type.
A concrete numerical test of the zeta corollary at small integer values of $u$ and $v$ would either exhibit the analytic continuation or locate the true domain of the identity, and this single test would decide all the specialised corollaries at once.

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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

First thing to know: the main theorem isn't proven and, as written, isn't true. The proof breaks at Equation (10). The author differentiates the Mellin-Barnes integral I_n(x), gets a Bessel expression, and integrates back from 0 to x. That step requires I_n(0+)=0 and a convergent t-integral. For Re(u)>k, which is exactly the theorem's stated domain, the t-integral diverges at 0 (the integrand behaves like t^{k-1-u}) and I_n(0+) is nonzero (the Mellin-Barnes poles at u-k-m give x^{k-u+m} terms). So (10) misses a constant and the final identity is not even a statement about ordinary absolutely convergent integrals. The zeta corollary makes this immediate: for k=1/2 the integrand is ~ t^{-1/2-u/2}, so every integral in Corollary 4.12 diverges for Re(u)>1. Same in the Ramanujan case, where Re(u)>13 but the integrand behaves like t^{11-u}. This is a load-bearing flaw, not a domain-optimization issue.

What is legitimately new: the paper aims at a unified Wilton-type product formula for all Hecke series, which I don't think exists in the literature. The applications list is ambitious, and the paper is clearly organized, with honest references to Banerjee et al. and earlier work. The underlying method—Riesz summation, residue passage, Meijer G-function manipulation—is a reasonable way to attack the problem. So credit where earned: the idea is natural and the exposition is readable.

Soft spots beyond the main error: the contour condition in the proof of Theorem 1.3 appears to be written with a max where a min is needed, and there are some character typos in the Dirichlet L-function section (χ vs χ-bar, and the second term in Corollaries 4.10 and 4.11 has the wrong Gauss sum factor). Those are minor compared to the divergence.

Who this is for: a reader who wants a survey of what a Wilton-type product identity could look like for Hecke series might glance at it, but anyone relying on the formulas will be misled. The paper deserves a serious referee only in the sense that the error is real and a referee could give the author a clear path to fix it: add the missing constant, restrict the domain to where the integrals converge, or replace the diverging integrals by regularized values. As it stands, I would not accept or cite it.

If this lands on my desk, I would not desk-reject without a second opinion, because the author is working on a real problem and the mistake is common. But I would expect rejection or a major revision before anything becomes publishable.

Referee Report

2 major / 4 minor

Summary. The paper proposes a general product identity for Dirichlet series satisfying Hecke's functional equation. Theorem 1.3 expresses phi(u)psi(v), for Re(u), Re(v) > max(c+1,k), as the sum of two residue terms minus two infinite series of Bessel-function integrals over [0,1]. The proof uses a Perron/Riesz operator F_a, the Hecke functional equation, a Mellin-Barnes/Meijer G representation, and a contour shift in the spirit of prior work by Banerjee-Mehta and Banerjee-Chakraborty-Hoque. The remaining sections derive corollaries for Hecke series, modular-form L-functions, Ramanujan's tau L-function, Epstein zeta functions, Dedekind zeta functions, Dirichlet L-functions, and a four-term identity for the Riemann zeta function. The paper is clearly organized and the intended scope is ambitious, but the central proof contains a critical invalid integration step.

Significance. If Theorem 1.3 were correct, the paper would provide a useful unified Wilton-type product formula covering many zeta and L-functions, and the breadth of applications is a genuine strength. The manuscript also follows established techniques rather than introducing ad hoc assumptions; there is no evidence of parameter fitting or circular reasoning. However, the main theorem rests on an unjustified integration step, and several displayed corollaries contain integrals that diverge on their stated ranges. In its present form the paper does not establish its headline results, and the flaws are load-bearing rather than cosmetic.

major comments (2)

[§3, Eq. (10)] Equation (10) is obtained by integrating the expression for I'_n(x) from 0 to x. This step requires I_n(0+)=0 and also requires the resulting Bessel integral to converge near t=0. Both requirements fail on the theorem's stated domain. From the Mellin-Barnes representation (8), the poles of Gamma(k-u+z) are at z = u-k-m for m=0,1,...; when Re(u)>k these yield contributions of order x^{k-u+m} to I_n(x), so I_n(x) does not vanish as x tends to 0+ and the antiderivative is determined only up to a nonzero constant that Eq. (10) drops. Independently, using J_{k-1}(z) ~ z^{k-1} near z=0, the integrand in Eq. (10) behaves like t^{3k/2 - 1 - u}, so the integral diverges at 0 whenever Re(u) >= 3k/2; this case is included in the theorem's hypothesis in standard applications (for example k=12 and Re(u)>13 already allow Re(u)>=18 for Ramanujan's L-function, and k=1 with Re(u)>1 allows Re(u)>=3/2 for Dedekind zeta functions). Thus Eq. (10), and consequently Eq. (11) and Theorem 1.3, are not established, and the claimed identity is not a valid identity of ordinary absolutely convergent integrals on the stated region.
[§4.6, Corollary 4.12] Corollary 4.12 asserts, for Re(u), Re(v) > 1, an identity for zeta(u)zeta(v) with integrals whose integrand is t^{-1/4-u/2} J_{-1/2}(2 pi n sqrt(t)). Since J_{-1/2}(z) ~ sqrt(2/(pi z)) near z=0, the integrand is asymptotic to a constant times t^{-1/2-u/2} near t=0. For every Re(u) > 1 this has a nonintegrable singularity at t=0, so the displayed identity cannot hold as a statement about ordinary convergent integrals. The remark after Theorem 1.3 that analytic continuation may extend the validity of Eq. (3) does not resolve this problem, because the integrals themselves have no finite value on the stated domain.

minor comments (4)

[§3, proof of Theorem 1.3] The rectangle in the contour-shift argument is described as having vertices '-a-iT, b-it, b+iT, -a+it'; the mixed use of t and T appears to be a typo, and the second and fourth vertices should presumably be b-iT and -a+iT.
[References] References [4] and [5] are listed with the same arXiv identifier 1611.08693 but with different titles; this needs to be checked and corrected.
[§4.6, Corollary 4.12 and §4.4, Corollary 4.9] The exponents in several corollaries appear inconsistent with direct substitution into Theorem 1.3; for example, Corollary 4.9 uses t^{-u} where the general formula with k=1 would suggest t^{1/2-u}. Please clarify the variable changes or correct the displayed exponents.
[§4.4, after Corollary 4.9] The statement that Corollary 4.9 is 'a mere rearrangement' of results in [4,5,6] is not accompanied by a comparison; spelling out the relation would improve the paper's contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main identity is derived from Hecke's functional equation and standard Mellin–Barnes manipulations, without fitting parameters or assuming its conclusion.

full rationale

The load-bearing result, Theorem 1.3, is obtained by combining the external Hecke correspondence theorem (Theorem 1.2), the definition of the Mellin–Barnes/Meijer G integral, a Perron-like identity proved in Lemma 2.1 (via Nakajima's dissection, an external result), and Lemma 2.2, which is proved directly by Dirichlet convolution. No parameter is fitted to data and no quantity predicted in the corollaries is an input to the derivation renamed. Citations to [3,6] are contextual descriptions of analogous methods; the present proof restates and proves the needed identities rather than deferring the central step to a self-citation. The paper even explicitly notes that Corollary 4.9 is a rearrangement of earlier results, which is an honest acknowledgment of limited novelty for that corollary and does not make the main theorem circular. The skeptical objection concerning Equation (10), namely that integrating I'_n(x) from 0 to x requires I_n(0+)=0 and convergence of the t-integral, is a mathematical-correctness and domain question, not a circularity question: the derivation does not assume the product identity it is trying to prove. Accordingly, no circular step meeting the required evidentiary standard is present, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper rests on standard Hecke correspondence and Meijer G identities; no free parameters or invented entities are introduced. The principal unstated assumption is that the contour manipulations and the integration from 0 to x are valid, which fails for Re(u)>=k.

assumptions (4)

domain assumption Hecke's correspondence theorem (Theorem 1.2) supplies the functional equation and the residue structure for the Dirichlet series.
The proof uses the equivalence between modular relations and functional equations to transform the contour integral. This is an external theorem taken as granted.
standard math Meijer G-function special case: G^{1,0}_{0,2}(;a,b|z) = z^{(a+b)/2} J_{a-b}(2 sqrt(z)).
This identity is used in Equation (8) and the subsequent differentiation to obtain the Bessel integrals. It is a known representation, but the paper applies it without proof.
standard math Nakajima dissection: sum_{m,n} = sum_m sum'_{n<=m} + sum_n sum'_{m<=n}.
This identity is used in Lemma 2.1 to derive the Perron-type formula. It is a standard double-counting identity.
domain assumption The contour shift and interchange of sums and integrals are justified by absolute convergence, relying on the boundedness in vertical strips from Theorem 1.2.
The proof does not verify all growth conditions explicitly, especially near the integration limit t=0, where the claimed integrability fails for Re(u)>=k.

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Cite this review

Pith. "Pith review of A Product Identity For Dirichlet Series Satisfying Hecke's Functional Equation." pith.science (2026). https://pith.science/paper/GMFQF7ML

@misc{pith2026250414551,
  author       = {Pith},
  title        = {Pith review of: A Product Identity For Dirichlet Series Satisfying Hecke's Functional Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMFQF7ML}},
  note         = {Machine review of arXiv:2504.14551}
}

abstract

In this paper, we give an analogue of Wilton's product formula for Dirichlet series that satisfy Hecke's functional equation. We apply our results to obtain identities for Hecke series, L-functions associated to modular forms, Ramanujan's L-function, Epstein zeta functions, Dedekind zeta functions of imaginary quadratic fields and Dirichlet L-functions. A $4$-term product identity for Riemann zeta function is also given.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

T. Apostol. Modular Functions and Dirichlet Series in Number Theory . Graduate Texts in Mathematics. Springer New York, 2012

work page 2012
[2]

Apostol and A

T. Apostol and A. Sklar. The approximate functional equation o f hecke’s dirichlet series. Transactions of the American Mathematical Society , 86(2):446–462, 1957

work page 1957
[3]

Banerjee and J

D. Banerjee and J. Mehta. Linearized product of two Riemann ze ta functions. Proceedings of the Japan Academy, Series A, Mathematical Sciences , 90(8):123 – 126, 2014

work page 2014
[5]

An analogue of Wilton's formula and values of Dedekind zeta functions

S. Banerjee, A. Hoque, and K. Chakraborty. On the product o f dedekind zeta functions. arXiv preprint arXiv:1611.08693, 2016

work page Pith review arXiv 2016
[6]

Banerjee, K

S. Banerjee, K. Chakraborty, and A. Hoque. An analogue of wilt on’s formula and values of dedekind zeta functions. Journal of Mathematical Analysis and Applications , 495(1):124675, 2021

work page 2021
[7]

Bateman and A

H. Bateman and A. Erd´ elyi. Higher transcendental functions, volume ii. Bateman Manuscript Project) Mc Graw-Hill Book Company , 1953

work page 1953
[8]

B. C. Berndt and M. I. Knopp. Hecke’s theory of modular forms and Dirichlet series , volume 5. World Scientiﬁc, 2008

work page 2008
[9]

Chandrasekharan and R

K. Chandrasekharan and R. Narasimhan. The approximate func tional equation for a class of zeta- functions. Mathematische Annalen , 152(1):30–64, 1963. 14

work page 1963

Show all 15 references

[10]

Davenport

H. Davenport. Multiplicative number theory , volume 74. Springer Science & Business Media, 2013

2013
[11]

P. Epstein. Zur theorie allgemeiner zetafunctionen. Mathematische Annalen , 56(4):615–644, 1903

1903
[12]

Hardy and J

G. Hardy and J. Littlewood. The approximate functional equat ions for ζ(s) and ζ2(s). Proceedings of the London Mathematical Society , 2(1):81–97, 1929

1929
[13]

G. H. Hardy and M. Riesz. The general theory of Dirichlet’s series . Courier Corporation, 2013

2013
[14]

A. F. Lavrik. Approximate functional equations for dirichlet fu nctions. Mathematics of the USSR- Izvestiya, 2(1):129, 1968

1968
[15]

Nakajima

M. Nakajima. A new expression for the product of the two Dirich let series I. Proceedings of the Japan Academy, Series A, Mathematical Sciences , 79(2):19 – 22, 2003

2003
[16]

J. Wilton. An approximate functional equation for the product of two ζ-functions. Proceedings of the London Mathematical Society , 2(1):11–17, 1930. 15

1930

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

T. Apostol. Modular Functions and Dirichlet Series in Number Theory . Graduate Texts in Mathematics. Springer New York, 2012

work page 2012

[2] [2]

Apostol and A

T. Apostol and A. Sklar. The approximate functional equation o f hecke’s dirichlet series. Transactions of the American Mathematical Society , 86(2):446–462, 1957

work page 1957

[3] [3]

Banerjee and J

D. Banerjee and J. Mehta. Linearized product of two Riemann ze ta functions. Proceedings of the Japan Academy, Series A, Mathematical Sciences , 90(8):123 – 126, 2014

work page 2014

[4] [5]

An analogue of Wilton's formula and values of Dedekind zeta functions

S. Banerjee, A. Hoque, and K. Chakraborty. On the product o f dedekind zeta functions. arXiv preprint arXiv:1611.08693, 2016

work page Pith review arXiv 2016

[5] [6]

Banerjee, K

S. Banerjee, K. Chakraborty, and A. Hoque. An analogue of wilt on’s formula and values of dedekind zeta functions. Journal of Mathematical Analysis and Applications , 495(1):124675, 2021

work page 2021

[6] [7]

Bateman and A

H. Bateman and A. Erd´ elyi. Higher transcendental functions, volume ii. Bateman Manuscript Project) Mc Graw-Hill Book Company , 1953

work page 1953

[7] [8]

B. C. Berndt and M. I. Knopp. Hecke’s theory of modular forms and Dirichlet series , volume 5. World Scientiﬁc, 2008

work page 2008

[8] [9]

Chandrasekharan and R

K. Chandrasekharan and R. Narasimhan. The approximate func tional equation for a class of zeta- functions. Mathematische Annalen , 152(1):30–64, 1963. 14

work page 1963