We are concerned here with Dirichlet series f(s) =3D 1 + Sigma(infinity)(n=3D2) c(n)/n(s) which satisfy a function equation similar to that of the Riemann zeta function, typically of the form f(s) =3D 2(s)q(1/2-s)pi(s-1)Gamma(1 - s) (sin pi/2(s + kappa)) f(1 - s), but for which the Riemann hypothesis is false. Indeed we show that the zeros of such functions are ubiquitous in the complex plane. (C) 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

In this paper, we are concerned with the representation of large positive integers as the sum of a fixed number of prime numbers taken from Beatty sequences. We are able to avoid any conditions on the Beatty sequences that they be of finite type which have been required hitherto. We use a form of the Hardy-Littlewood-Vinogradov method, but the proofs are quite delicate.

In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarnik type theorems established in Beresnevich et al. (Ann Math 166(2):367-426, 2007) and Vaughan and Velani (Invent Math 166:103-124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of 'shifted' rational points near planar curves.

There is, apparently, a persistent belief that in the current state of knowledge it is not possible to obtain an asymptotic formula for the number of partitions of a number n into primes when n is large. In this paper such a formula is obtained. Since the distribution of primes can only be described accurately by the use of the logarithmic integral and a sum over zeros of the Riemann zeta-function one cannot expect the main term to involve only elementary functions. However the formula obtained, when n is replaced by a real variable, is in C(infinity) and is readily seen to be monotonic.

The convergence theory for the set of simultaneously psi-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in [1] and thereby completes the general metric theory for planar curves.