In this paper, we initiate a rigorous and detailed study of the smallest denominator function and the Riemann function. For this, we rst establish some basic facts about real numbers and the divisibility of integers.

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula ...

The theory of S2(δ) family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The δ deformation of the xi function is formulated in terms of the S2(δ) distribution and shown to satisfy Riemann’s functional equation. Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are form...

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A key result provides a basis for generalizing the important Riemann-Siegel integral formula.

In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of ζ(ρ) where ζ(s) is the Riemann zeta function and ρ is a non-trivial zero of the Riemann zeta function.