A singular series average and Goldbach numbers in short intervals

Abstract

A SINGULAR SERIES AVERAGE AND GOLDBACH NUMBERS INSHORT INTERVALSbyA. LANGUASCO1. IntroductionThis is the last preprint. The final paper appeared in Acta Arith., 83 (1998),171-179.The Goldbach conjecture asserts that every even integer greater than two can be written as asum of two primes. We will call G-numbers the even numbers satisfying Goldbach’s conjecture.LetX beasufficientlylargeparameter. OuraimhereistostudyG-numbersinshortintervals.The first unconditional result of this kind was obtained by Ramachandra [8] who proved thatthere is a positive proportion of G-numbers in every interval [X,X +H] for H ≥ X