The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane $$\{s\in {\mathbb {C}}:\mathrm{Re}\, s>1\}$$ { s ∈ C : Re s > 1 } . In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line $$\mathrm{Re}\, s=1$$ Re s = 1 . In particular, regarding the Riemann zeta function $$\zeta (s)$$ ζ ( s ) , for every $$\sigma _0>1$$ σ 0 > 1 we can assure the existence of a relatively dense set of real numbers $$\{t_m\}_{m\ge 1}$$ { t m } m ≥ 1 such that the parametrized curve traced by the points $$(\mathrm{Re} (\zeta (\sigma +it_m)),\mathrm{Im}(\zeta (\sigma +it_m)))$$ ( Re ( ζ ( σ + i t m ) ) , Im ( ζ ( σ + i t m ) ) ) , with $$\sigma \in (1,\sigma _0)$$ σ ∈ ( 1 , σ 0 ) , makes a prefixed finite number of turns around the origin.

An Analytical and Numerical Detour for the Riemann Hypothesis

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

Bounding the log-derivative of the zeta-function

Assuming the Riemann hypothesis we establish explicit bounds for the modulus of the log-derivative of Riemann’s zeta-function in the critical strip.

Discrete Approximation by a Dirichlet Series Connected to the Riemann Zeta-Function

In the paper, a Dirichlet series ζuN(s) whose shifts ζuN(s+ikh), k=0,1,⋯, h>0, approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence uN→∞ and uN≪N2 as N→∞.