Zeta-functions of harmonic theta-series and prime numbers

Abstract

Speaking on rational prime numbers in various arithmetical sequences, it may be noted that no essential progress have been achieved for more than century and a half since famous Dirichlet theorem on prime numbers in arithmetic progressions (1837). Absolutely mystical is still the question on prime numbers in quadratic sequences, i.e., on prime numbers of the form an+bn+c, where a, b, c are rational coprime integers, and d = b − 4ac is not a rational square. The situation is not changing despite of considerable progress of the algebraic-analytical theory of integral quadratic forms reached after Dirichlet. Our purpose here is to draw the attention of numbertheorist to some analytical aspects of the theory of quadratic forms possibly related to the problem. In order to be more concrete, let us start from the the celebrated problem on prime numbers of the form 1+n. It is well known that the problem is closely related to reductions prime modules of certain elliptic curves with complex multiplications by Gauss integers a+ √ −1b, say, the curve y = x(x − 1). (1.1)