The Schinzel hypothesis is a conjecture on prime values in polynomials. We prove it by Golomb's method.

The Bateman-Horn conjecture is a conjecture on prime values in polynomials. We prove it by Golomb's method.

Pausinger recently investigated a special determinant involving prime numbers. In this short note we point out that this type of determinants was already known in linear algebra and its computation is unrelated to prime numbers.

In this paper we first give a new formula of the Liouville function and then by using the method for proving the Bateman-Horn conjecture, we give a parallel proof of the Chowla conjecture.

Let $ \xi(z) $ be the Riemann xi function. In a previous paper we prove the boundedness of coefficients of the power series expansion of $ \xi'(1/z)/\xi(1/z) $ and thus give a proof of the Riemann hypothesis. In this paper we generalize the method there to the study of the extended Riemann hypothesis for general number fields.

Let ξ(z) be the Riemann xi function. We prove the boundedness of coefficients of the power series expansion of ξ′(1/z)/ξ(1/z). By an observation of Keiper this implies that the Riemann hypothesis is true.