{"id":"97c631c1-8397-4ae9-b07c-37d597dd8ad0","arxiv_id":"2504.14551","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A proposed analogue of Wilton's product formula for Hecke series is invalid as stated because its Bessel integrals diverge on the claimed range Re(u)>k.","lead":"The paper claims a general product identity for Dirichlet series satisfying Hecke's functional equation, generalizing Wilton's formula. The main theorem's proof appears to contain a domain error that makes the Bessel integrals diverge exactly in the stated region.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (10) integrates I'_n(x) from 0 to x without checking I_n(0+)=0; for Re(u)>k the omitted constant is nonzero, so the final Bessel-integral identity fails on the stated domain and Corollary 4.12's integrals diverge for all Re(u)>1.","rationale":"The reader identifies the same load-bearing step: Eq. (10) integrates the derivative I'_n(x) from 0 to x, which requires I_n(0+)=0, and this fails for Re(u)>k. I agree that the proof is invalid there and that the theorem as stated cannot hold. The reader's stated reason is slightly imprecise: the convergence of the final t-integral at t=0 actually persists for Re(u)<3k/2, while the failure of I_n(0)=0 is what invalidates Eq. (10) for all Re(u)>k. Both effects damage the theorem, and the zeta corollary provides an unconditional, easily checked counterexample because its integrals diverge for every u>1. The corollaries inherit the flaw, and the paper offers no independent verification, machine-checked proof, or alternative derivation. Hence the reader's REJECT verdict is correct and no verdict adjustment is needed. The only secondary issue is the b<max(...) condition, which should be a min, but the primary failure is already decisive.","tokens_in":12870,"tokens_out":10008,"duration_ms":88994,"concrete_test":"In Corollary 4.12 take n=1 and u=2. The first summand contains integral_0^1 t^{-1/4-1} J_{-1/2}(2 pi sqrt(t)) dt. Using J_{-1/2}(z) = sqrt(2/(pi z)) cos z, the integrand is (1/pi) t^{-3/2} cos(2 pi sqrt(t)). On intervals where cos(2 pi sqrt(t)) >= 1/2, the integrand is comparable to t^{-3/2}, so the improper integral diverges. Thus the asserted identity cannot hold for u=2. For the main theorem, independently compute I_n(epsilon) by closing the defining Mellin-Barnes contour to the left when Re(u)>k and observe the leading residue term C epsilon^{k-u}, confirming that I_n(0+) is not zero and Eq. (10) is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 rests on Eq. (10), which is obtained by differentiating the Mellin-Barnes integral I_n(x) and then integrating the result from 0 to x. This step requires both convergence of the t-integral at t=0 and I_n(0+)=0. Both requirements fail on the theorem's domain. The poles of Gamma(k-u+z) in the definition of I_n(x) lie at z=u-k-m. When Re(u)>k, these poles give contributions to I_n(x) of order x^{k-u+m} as x tends to 0, so I_n(x) does not tend to 0; the antiderivative is determined only up to a nonzero constant that Eq. (10) drops. Moreover, the t-integral itself diverges at 0 whenever Re(u) >= 3k/2; for example, in the Ramanujan case k=12, this is Re(u)>=18, well inside the claimed Re(u)>13. The zeta corollary makes the failure immediate: with k=1/2, the integrand t^{-1/4-u/2}J_{-1/2}(2 pi sqrt(nt)) behaves like (1/pi) t^{-1/2-u/2}, so for every Re(u)>1 the integral over [0,1] diverges and the displayed identity is not a valid statement about ordinary absolutely convergent integrals. This is not a mere domain-optimization issue; the central identity as written is false on a substantial part of its stated domain.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13191,"tokens_out":20543,"duration_ms":174279,"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":[{"comment":"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.","section":"§3, Eq. (10)"},{"comment":"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.","section":"§4.6, Corollary 4.12"}],"minor_comments":[{"comment":"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.","section":"§3, proof of Theorem 1.3"},{"comment":"References [4] and [5] are listed with the same arXiv identifier 1611.08693 but with different titles; this needs to be checked and corrected.","section":"References"},{"comment":"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.","section":"§4.6, Corollary 4.12 and §4.4, Corollary 4.9"},{"comment":"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.","section":"§4.4, after Corollary 4.9"}],"recommendation":"reject","confidential_remarks":"The central identity is not proven and several corollaries are not meaningful as stated because the displayed integrals diverge on the claimed ranges. I see no indication of bad faith or circular reasoning; the error is a technical integration-domain mistake that invalidates the main theorem. Because a repair would require changing the form of the theorem and reworking essentially all applications, rejection is appropriate, although the derivation up to Eq. (9) may contain useful ideas for a future corrected version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nWho 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.\n\nIf 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.","headline":"The central product identity is false as stated: the Bessel integrals diverge on the claimed domain and Equation (10) silently drops a nonzero constant.","tokens_in":13721,"tokens_out":4072,"would_cite":false,"duration_ms":37638,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Hecke-series Dirichlet functions, a product of two values equals residue terms plus Bessel-integral sums.","keywords":["Wilton's formula","Hecke series","Hecke functional equation","Dirichlet series","Meijer G-function","Bessel integrals","L-functions","Riemann zeta function"],"falsifier":"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.","tokens_in":12654,"feed_emoji":"📐","tokens_out":16103,"duration_ms":130997,"temperature":0.7,"pith_summary":"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.","feed_headline":"Wilton-style product formula now spans all Hecke Dirichlet series","feed_subtitle":"One identity covers zeta, modular L-functions, Epstein zeta, and Dirichlet L-functions.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\varphi(u)^2=\\frac{2}{u-k}\\mathrm{Res}\\,\\varphi(k)\\varphi(2u-k)-\\frac{4\\pi\\gamma}{\\lambda}\\sum_n \\sigma_{\\alpha,k-2u}(n)n^{1-k/2}\\int_0^1 t^{k/2-1/2-u}J_{k-1}(4\\pi\\sqrt{nt}/\\lambda)\\,dt$."],"supporting_citations":[{"why":"Wilton's original approximate functional equation for the product of two zeta functions, the identity this paper generalises.","marker":"[16]"},{"why":"Supplies the modern statement of Wilton's formula for the Riemann zeta function and the Perron/Riesz-summation technique used in the proof.","marker":"[3]"},{"why":"Earlier treatment of approximate functional equations for Hecke's Dirichlet series, the framework being extended to full product identities.","marker":"[2]"},{"why":"Provides Hecke's correspondence theorem and the definition of Hecke series, which set up the hypotheses of Theorem 1.3.","marker":"[8]"},{"why":"Defines the Meijer G-function and its Bessel-function special case, the kernel that carries the contour integral to the final formula.","marker":"[7]"},{"why":"The double-sum dissection used in Lemma 2.1 to split the product of the two Dirichlet series.","marker":"[15]"},{"why":"Earlier analogue of Wilton's formula for Dedekind zeta functions; the paper's imaginary-quadratic-field corollary is a rearrangement of this result.","marker":"[6]"}],"fun_headline_variants":["Hecke Dirichlet series gain Wilton-style product identity","Product formula for Hecke series and zeta functions","Unified product identity for Hecke and L-functions","Wilton analogue covers zeta, Epstein, and Dirichlet L-functions","New identity links Hecke, zeta, and modular L-functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hecke Dirichlet series gain Wilton-style product identity","Product formula for Hecke series and zeta functions","Unified product identity for Hecke and L-functions","Wilton analogue covers zeta, Epstein, and Dirichlet L-functions","New identity links Hecke, zeta, and modular L-functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000296,"raw_usage":{"total_tokens":1737,"prompt_tokens":986,"completion_tokens":751,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":664}},"tokens_in":602,"tokens_out":751,"duration_ms":6201,"temperature":1.0,"reasoning_tokens":664,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:49:35.885903+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Hecke's correspondence theorem and the definition of Hecke series, which set up the hypotheses of Theorem 1.3."},{"cited_title":"Apostol and A","cited_arxiv_id":null,"evidence_quote":"Earlier treatment of approximate functional equations for Hecke's Dirichlet series, the framework being extended to full product identities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Wilton's original approximate functional equation for the product of two zeta functions, the identity this paper generalises."},{"cited_title":"Banerjee and J","cited_arxiv_id":null,"evidence_quote":"Supplies the modern statement of Wilton's formula for the Riemann zeta function and the Perron/Riesz-summation technique used in the proof."},{"cited_title":"Bateman and A","cited_arxiv_id":null,"evidence_quote":"Defines the Meijer G-function and its Bessel-function special case, the kernel that carries the contour integral to the final formula."},{"cited_title":"Nakajima","cited_arxiv_id":null,"evidence_quote":"The double-sum dissection used in Lemma 2.1 to split the product of the two Dirichlet series."},{"cited_title":"Banerjee, K","cited_arxiv_id":null,"evidence_quote":"Earlier analogue of Wilton's formula for Dedekind zeta functions; the paper's imaginary-quadratic-field corollary is a rearrangement of this result."}],"review_version":1}