Let Λ be the von Mangoldt function andR(n)= ∑ h+k=nΛ(h)Λ(k) be the counting function for the Goldbach numbers. Let N ≥ 2 and assume that the Riemann Hypothesis holds. We prove that N ∑ n=1 R(n) = N2 2 − 2 ∑ ρ Nρ+1 ρ(ρ+ 1) +O(N log N), where ρ = 1/2+iγ runs over the non-trivial zeros of the Riemann zeta-function ζ(s). This improves a recent result by Bhowmik and Schlage-Puchta.

The classical Green’s function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green’s function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analo...

Riemann’s way to calculate his zeta function on the critical line was based on an application of his saddle-point technique for approximating integrals that seems astonishing even today. His contour integral for the remainder in the Dirichlet series for the zeta function involved not an isolated saddle, nor a saddle near a pole or an end-point or several coalescing saddles, but the configuratio...

This talk will come in four parts. First, I will introduce the Riemann hypothesis (RH) as it was first introduced, using the Riemann zeta function, and discuss briefly its connection to number theory and prime numbers. Second, I will make this connection explicit by discussion a famous problem equivalent to RH. Third, I will examine the generalised Riemann hypothesis (GRH) and, finally, will li...

1. Assume everything you know from the notes on Riemann-Stieltjes integration. (a) Show that if g : [0, 1]→ R is Riemann-integrable on [0, 1] and φ : [a, b]→ R is continuous on an interval [a, b] ⊇ g[[0, 1]], then φ ◦ g is Riemann-integrable on [0, 1]. (b) Assume that f : [0, 1]→ R is a function for which f ′(t) exists at each t ∈ [0, 1] (one-sidedly at endpoints) and that the Riemann integral ...

“Fuzzy” models and 802.11 mesh networks have garnered improbable interest from both computational biologists and experts in the last several years. Given the current status of real-time epistemologies, futurists daringly desire the construction of DHCP, which embodies the practical principles of networking. In order to accomplish this ambition, we introduce new concurrent methodologies (Krang),...

The Riemann zeta function is a meromorphic functionon the whole complex plane. It has infinitely many zeros and aunique pole at s = 1. Those zeros at s = −2,−4,−6, . . . areknown as trivial zeros. The Riemann hypothesis, conjectured byBernhard Riemann in 1859, claims that all non-trivial zeros of ζ(s)lie on the line R(s) =12 . The density hypothesis is a conjecturede...

We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call “value recycling”...

We prove a variant of Connes’s trace formula and show how it can be used to give a new proof of the Riemann hypothesis for L-functions with Größencharacter for function fields.

Turing encountered the Riemann zeta function as a student, and developed a life-long fascination with it. Though his research in this area was not a major thrust of his career, he did make a number of pioneering contributions. Most have now been superseded by later work, but one technique that he introduced is still a standard tool in the computational analysis of the zeta and related functions...