A quadratic critical-value conjecture for the fifth Bessel moment
Abstract
We conjecture an explicit evaluation of the pure fifth Bessel moment ∫0∞ K0(t)5 dt as a quadratic expression in the critical value L(f,2) of the weight-three, level-60 newform f (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in ℚ(√5) and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or L-values, bound the absolute discrepancy by 10−358. We prove three exact modular identities for f: the Petersson-norm formula ⟨ f,f⟩60 = (3(5−√5))/2π4|L(f,2)|2, the coefficient-conjugation relation L(fσ,2) = κL(f,2) with explicit κ∈ ℚ(√5,i), and the twisted symmetric-square evaluation L(χ−4Sym2 f,2) = √(15) π2 ⟨ f,f⟩60, together with L(χ−4Sym2 f,3) = π4⟨ f,f⟩60/8, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture D5,odd = (3√(15)(5−√5))/2|L(f,2)|2 is equivalent to the symmetric-square conjecture D5,odd = π2 L(χ−4Sym2 f,2) of Lim, Tu and Yu, while the exact relation D5,even = π2 D5,odd/(2√(15)) follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.
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BibTeX
@misc{matuzas2026quadratic,
title = {{A quadratic critical-value conjecture for the fifth Bessel moment}},
author = {Jonas Matuzas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22318},
eprint = {2609.22318},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22318},
}
RIS
TY - UNPB
TI - A quadratic critical-value conjecture for the fifth Bessel moment
AU - Jonas Matuzas
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.22318
UR - https://arxiv.org/abs/2609.22318
AB - We conjecture an explicit evaluation of the pure fifth Bessel moment ∫_0^∞ K_0(t)^5 dt as a quadratic expression in the critical value L(f,2) of the weight-three, level-60 newform f (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in ℚ(√5) and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or L-values, bound the absolute discrepancy by 10^−358. We prove three exact modular identities for f: the Petersson-norm formula ⟨ f,f⟩_60 = (3(5−√5))/2π^4|L(f,2)|^2, the coefficient-conjugation relation L(f^σ,2) = κL(f,2) with explicit κ∈ ℚ(√5,i), and the twisted symmetric-square evaluation L(χ_−4Sym^2 f,2) = √(15) π^2 ⟨ f,f⟩_60, together with L(χ_−4Sym^2 f,3) = π^4⟨ f,f⟩_60/8, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture D_5,odd = (3√(15)(5−√5))/2|L(f,2)|^2 is equivalent to the symmetric-square conjecture D_5,odd = π^2 L(χ_−4Sym^2 f,2) of Lim, Tu and Yu, while the exact relation D_5,even = π^2 D_5,odd/(2√(15)) follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.
ER -