Eventual Sign-Regularity of Nuttall Cumulants: Finiteness in Conjecture 1

Abstract

John Nuttall introduced a sequence of cumulants Ψ𝑚 associated with the Riemann 𝜉-kernel Φ andconjectured that, for every fixed order 𝑟 > 1, there is a finite least index 𝑚(𝑟) for which the additiveHankel kernel 𝐾𝑚(𝑢, 𝑣) = Ψ𝑚(𝑢 + 𝑣) is sign-reverse regular of order 𝑟. We prove the finiteness assertion𝑚(𝑟) < ∞ for every fixed 𝑟. The proof combines an exact first-theta-mode Wronskian identity, a right-tailsign-regularity argument, an Andréief representation for Toeplitz-type moment determinants, and alocalization estimate for high moments. The only region where the derivative determinant can have thewrong sign forces one integration variable to remain bounded; this loses one full logarithmic momentfactor compared with an explicitly constructed positive box in the theta tail. Quantitative first-modedominance on that box then yields eventual positivity of the moment determinants, uniformly in thetranslation parameter. An exact Toeplitz–Wronskian identity transfers this positivity to the cumulants,and the Nuttall–Karlin differential criterion gives full sign-regularity. The result is qualitative: no usefulupper bound on 𝑚(𝑟) uniform in 𝑟 is obtained, and no claim concerning the Riemann hypothesis itselfis made.