Cascade Prime Entry Periods in Eisenstein Series via Von Staudt–Clausen

We prove that the normalization constant of the weight-10 Eisenstein series E10 is −264 = −24 × 11, where 24 = Im((2+i)⁴) and 11 = Im((2+i)³). The Bernoulli number B10 = 5/66 encodes the seed norm N(2+i) = 5 in its numerator. Cascade primes enter Bernoulli denominators at cascade-determined periods via Von Staudt–Clausen: p = 5 at period 4 = |Z4|, p = 17 at period 16 = 4², p = 137 at period 136 = 8 × 17. The nexus value 264 = 24 × 11 appears simultaneously in the hyperdeterminant I4 (Paper 13) and in E10, connecting Gaussian lattice arithmetic to modular form theory.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-31
DOI
https://doi.org/10.5281/zenodo.23843130
Primary Topic
Advanced Mathematical Identities
Type
preprint
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preprint

Cascade Prime Entry Periods in Eisenstein Series via Von Staudt–Clausen

Robert A. Kenney
Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Identities
preprint

Cascade Prime Entry Periods in Eisenstein Series via Von Staudt–Clausen

Robert A. Kenney
preprint en

Abstract

We prove that the normalization constant of the weight-10 Eisenstein series E10 is −264 = −24 × 11, where 24 = Im((2+i)⁴) and 11 = Im((2+i)³). The Bernoulli number B10 = 5/66 encodes the seed norm N(2+i) = 5 in its numerator. Cascade primes enter Bernoulli denominators at cascade-determined periods via Von Staudt–Clausen: p = 5 at period 4 = |Z4|, p = 17 at period 16 = 4², p = 137 at period 136 = 8 × 17. The nexus value 264 = 24 × 11 appears simultaneously in the hyperdeterminant I4 (Paper 13) and in E10, connecting Gaussian lattice arithmetic to modular form theory.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Identities
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Cascade Prime Entry Periods in Eisenstein Series via Von Staudt–Clausen — Robert A. Kenney · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS