Towards a Statistical Proof of the Riemann Hypothesis
نویسنده
چکیده
Using the ζ functional equation and the Hadamard product, an analytical expression for the sum of the reciprocal of the ζ zeros is established. We then demonstrate that on the critical line, |ζ| is convex, and that in the region 0 < <(s) ≤ 0.5, |ζ| has a negative slope, given the RH. In each case, analytical formulae are established, and numerical examples are presented to validate these formulae.
منابع مشابه
Local Extrema of the Ξ(t) Function and The Riemann Hypothesis
In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function Ξ(t) = ξ( 1 2 + it). First, we prove that positivity of all local maxima and negativity of all local minima of Ξ(t) form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of Ξ(t) is a ...
متن کاملGeneralized Riemann Hypothesis
(Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems ([BRW], [SA])...
متن کاملFinal steps towards a proof of the Riemann hypothesis
A proof of the Riemann’s hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D in one dimension, and their respective eigenfunctions ψs(t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fu...
متن کاملOn the Theory of Zeta-functions and L-functions
In this thesis we provide a body of knowledge that concerns Riemann zeta-function and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet’s L-function. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we ...
متن کاملProof of Riemann Hypothesis
Hilbert-Polya conjecture and Riemann hypothesis are proven. The construction of Hilbert-Polya operator is inspired by the conviction that Riemann Zeta function is associated with a physical system allowing superconformal transformations as its symmetries. The proof as such is elementary involving only basic facts about the theory of Hilbert space operators and complex analysis.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009