There Are Infinitely Many Prime Twins

Here we will show that T (s)− B2(s − 1) −1 can be continuously extended onto the line Re s = 1, (Th.1). This enables application of the powerful complex Tauberian theorem of Wiener and Ikehara (1931) and then almost immediately yields our main result, (Th.2). The ideas basic to our proof of Th.1 are plain: We are striving to exhibit a relation between T (s) and the Riemann Zeta function ζ(s); motivated by a proof of the Prime Number Theorem along the above lines [where T (s) is replaced by [log ζ(s)] and only ζ(s) 6= 0 on Re s = 1 needs to be known]. Now, T (s) can be obtained by a “kind of sieving” from the Dirichlet series of [log ζ(s)], which leads to the arithmetic formula (2) in Lemma 1 for the “characteristic” function Λ(n− 1)Λ(n+ 1) of the prime powers with difference 2. This important formula leads to a representation of T (s) by a double series over all odd squarefree m in Z and over all n in Z satisfying the quadratic congruence 4n ≡ 1 mod m. The number of such n in any interval of length m is a known multiplicative function of m, which heuristically leads to the above singular term for T (s) with, remarkably, the same constantB2 as made plausible by the classical circle method of Hardy & Littlewood. But splitting off such a “main term” and studying the remainder turned out be counterproductive. In this context a look at some papers of P. Turan [5], [6] is informative.