Abstract. For a prime polynomial f(T) ∈ Z[T], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f(T) ∈ κ[u][T], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ[u]) is not generally true when f(T)isapolynomial in T p (p the characteristic of κ). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values µ(f(g)) as g varies. We prove the surprising fact that this “Möbius average, ” which can be defined without reference to any conjectures, has a periodic behavior governed by the geometry of the plane curve f =0. The periodic Möbius average behavior implies in specific examples that a polynomial in κ[u][T] does not take prime values as often as analogies with Z[T] suggest, and it leads to a modified conjecture for how often prime values occur. 1.

The Wiener--Khintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a Wiener--Khintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PACS: 05.40+j; 02.30.Nw; 02.10.Lh Keywords: Twin primes; Ramanujan--Fourier series; Wiener--Khintchine formula 1. Introduction " The Wiener--Khintchine theorem states a relationship between two important characteristics of a random process: the power spectrum of the process and the correlation function of the process" [1]. One of the outstanding problems in number theory is the problem of prime pairs which asks how primes of the form p and p+h (where h is an even integer) are distributed. One immediately notes that this is a problem of #nding correlation between primes. We make two key observations. First of all there is an arithmetical function (a function de#ned on integers) which traps the...

Using Jiang function we prove Jiang prime k-tuple theorem. We prove that the Hardy-Littlewood prime k-tuple conjecture is false. Jiang prime k-tuple theorem can replace the Hardy-Littlewood prime k-tuple conjecture.

Using Jiang function we prove Jiang prime k-tuple theorem. We prove that the Hardy-Littlewood prime k-tuple conjecture is false. Jiang prime k-tuple theorem can replace the Hardy-Littlewood prime k-tuple conjecture.

Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be Ramanujan- Fourier series. 1 Introduction. The two well known methods in additive number theory are the circle method and the sieve method. The circle method is based on using a generating function (See Section 3) and noting along with Ramanujan and Hardy that the rational points on the circle contribute most and then through estimates showing that the contribution from the other points is