TITLE:
The Exponential Form of the Bernoulli Expansion of the Riemann Zeta Zero Condition and Perfect Numbers
AUTHORS:
Michael Mark Anthony
KEYWORDS:
Riemann Hypothesis, Bernoulli Numbers, Exponential Sums, Mellin Transform, von Staudt-Clausen Theorem, Dirichlet Eta Function, Riemann-Siegel Theta, Infinite Products
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.9,
September
2,
2026
ABSTRACT: This paper establishes theorems about the classical zero condition of the Riemann Zeta function—a resummation identity, exact boundary and factorization laws, and a conditioning analysis of its representations—and claims no new constraint on the location of the zeros. The zero condition of the Riemann Zeta function, expressed through a Bernoulli-Gamma expansion derived here in full from the Jensen integral form of
ζ(
s
)
, is shown to be the asymptotic shadow of a convergent Mellin integral whose kernel is the exponential generating function of the Bernoulli numbers,
x/
(
e
x
−1
)
. The integral evaluates in closed form, and the divergent expansion acquires an exact meaning: at every nontrivial zero
ρ
, its resummed value equals the rational form
ρ/
(
ρ−1
)
precisely. Three progressively more explicit exponential forms of the sum are derived, terminating in a convergent sum of pure exponentials
e
−ρlnm
whose vanishing is the zero condition itself. The asymptotic regime of the raw series is characterized exactly by the bound
|
(
ρ+1
)(
ρ+2
) |<
π
2
, a region containing no nontrivial zeros. A von Staudt-Clausen decomposition separates an integer component from a prime component with elementary closed form, and the conditioning of every representation at the zeros is measured, forming an exponential ladder
e
πτ
,
e
πτ/2
,
e
πτ/4
that terminates at exponent zero in the paired representation: the alternating sum, opened pair by pair through the product factorization of
e
ax
−
e
bx
, becomes an absolutely convergent sum of sinh atoms carried on the geometric-mean midpoints of consecutive integers, polynomially conditioned at every height. The atoms of the resulting factorizations own precisely the two edges of the critical strip—gap atoms the left edge, prime atoms the right edge—so that the interior zeros arise only as inter-pair interference of the midpoint carriers. The same atom shape uniformizes across the
L
-function family, separating the zero conditions of
ζ
and of the Dirichlet beta function into distinct product sums. Results are established as theorems with proof, as classical facts with citation, or, in a small number of explicitly labelled cases, as computational observations; the identities have additionally been confirmed numerically at the nontrivial zeros themselves.