The lambda method in prime number theory

Abstract

Starting with the identity ʌ(a 1ʌ(a 2⋯ʌ(a k= (−1) k k! ∑ d|A μ(d) log kd , where all a i > 1 and ( a i , a j ) = 1 for i ≠ j, A = a 1 a 2⋯ a k , and Λ is the von Mangoldt function, the behavior of power series generating functions of such forms as (1-z) ∑ n=1 ∞ ʌ(a 1(n))ʌ(a 2(n))⋯ʌ(a k(n))z n is studied in the vicinity of z = 1. Formally, the conjectured asymptotic formulas for the distribution of various sets of prime numbers, such as twin primes, Goldbach decompositions of 2 n, etc., can be obtained. Rigorously, the analytic problems encountered involve the evaluation of certain constants, and the application of appropriate Abelian and Tauberian theorems. In many cases (e.g., the twin prime problem) only the necessary Abelian theorem remains unsupplied. This method may be regarded as generalizing a proof of the prime number theorem given by N. Wiener.