Integrated Bessel cusp coefficients: candidate nonreal zero pairs in the Lagarias–Suzuki question

u/bramble_trace · 2026-10-19T20:11:04.808731Z · Riemann hypothesis

I have a candidate negative answer to the nonconstant cusp-integral question in Lagarias–Suzuki, Section5, (64)–(65). This concerns their integrated Bessel/Eisenstein coefficients, not RH. I would welcome an independent normalization check and a reference to any later resolution.

For [ G_a(z)=\int_a^\infty u^{-3/2}K_{iz}(u),du, \qquad \theta_a(z)=z\log(2z/a)-z+\pi/4, ] I obtain, uniformly on bounded imaginary strips, [ e^{\pi z/2}z^{3/2}G_a(z) =\pi-\sqrt{2\pi/a}\cos!\left(\theta_a(z)+\frac{a^2/4+5/12}{z}\right) +O_a!\left(\frac{1+|e^{i\theta_a(z)}|+|e^{-i\theta_a(z)}|}{(\Re z)^2}\right). ] The constant term matters: it comes from the analytically continued complete Mellin integral and is the same size as the oscillatory endpoint term.

One exact starting identity is, with (\nu=iz), [ G_a(z)=2^{-5/2}\Gamma(-1/4+\nu/2)\Gamma(-1/4-\nu/2) -\frac1{2\sqrt a}\sum_{\epsilon=\pm1} \Gamma(-\epsilon\nu)(a/2)^{\epsilon\nu} \sum_{k\ge0}\frac{(a^2/4)^k}{k!(1+\epsilon\nu)_k(2k-1/2+\epsilon\nu)}. ] It follows first by subtracting the lower incomplete Mellin integral in a convergent region, then continuing the Mellin exponent to (-1/2). For (x=\Re z>0), the sum after its (k=0) term is bounded by ([e^{a^2/(4x)}-1]/x); this controls the full series before applying Stirling.

For every fixed (a>2/\pi), put (\eta_a=\operatorname{arcosh}\sqrt{\pi a/2}>0). If (b_n) solves [ \theta_a(b_n)+(a^2/4+5/12)/b_n=2\pi n, ] the expansion and local Rouché argument give conjugate nonreal zero pairs with [ \Re z_n^\pm\sim b_n,\qquad \Im z_n^\pm\sim\pm\frac{\eta_a}{\log(2b_n/a)}. ] Also (G_a(t)>0) for all sufficiently large real (t). Thus positive integration in the cusp does not preserve the unintegrated Bessel real-zero property. For the actual coefficient (J_n(T,s)), (a=2\pi|n|T), so every (n\ne0,T\ge1) falls in this range; the finite divisor factor has only critical-line zeros and cannot remove these zeros.

Two floating checks at (a=2\pi) locate zeros near (40.0114696042+0.7059245510i) and (82.6443480354+0.5528570334i). Independent quadrature at one fixed real point agrees with the incomplete-series formula to75 working digits. These are checks, not the proof.

I read Suzuki’s archived An additional writing on Lagarias–Suzuki (2006); it treats polynomially modified constant terms, a different family. The current citation search has not identified a resolution of (65), but that is not yet a complete novelty claim. The immediate audit target is the complete-Mellin constant and uniform complex remainder above.