TITLE:
A Self-Referential Integral Equation for the Nontrivial Zeros of the Riemann Zeta Function
AUTHORS:
Michael Mark Anthony
KEYWORDS:
Riemann Hypothesis, Zeta Function, Non-Trivial Zeros, Critical Line, Perron, Fubini Theorem, Primes, Prime Counting, Bernoulli, Entire Functions, Von Staudt-Clausen
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.7,
July
24,
2026
ABSTRACT: Starting from Jensen’s integral representation of the Riemann zeta function, I derive, by a chain of convergent algebraic manipulations, a self-referential integral equation that a complex number F must satisfy in order for
ρ=
F(
s
)/
(
F(
s
)−1
)
to be a nontrivial zero of the zeta-function, ζ. The resulting equation, is an explicit kernel and a biconditional characterization of the F-image of the nontrivial zeros. From calculations, I show that the Riemann Hypothesis is equivalent to the statement |F| = 1 for every solution. The equation simplifies dramatically to a Dirichlet-like series of lower incomplete gamma functions evaluated on the imaginary axis. A new representation of the Jensen’s integral for the Zeta function is show to be reducible to a series relating to Von Staudt-Clausen theorem. These new relations are shown to have a direct bearing on the relationship of the non-trivial zeros of the Zeta function and the primes. I discuss the analytical constraints and outline what is needed to close the gap to a proof of the Riemann Hypothesis.