On the Distribution of M-tuples of B-numbers

Werner Georg,Nowak

In the classical sense, the set $B$ consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field $\Qi(i)$. G. J. Rieger (1965) and T. Cochrane. R. E. Dressler (1987) established bounds for the number of pairs $(n,n+h)$, resp., triples $(n,n+1,n+2)$ of $B$-numbers up to a large real parameter $x$. The present article generalizes these investigations into two directions: The result obtained deals with arbitrary $M$-tuples of arithmetic progressions of positive integers, excluding the trivial case that one of them is a constant multiple of one of the others. Furthermore, the estimate applies to the case of an arbitrary normal extension $K$ of the rational field instead of $\Qi(i)$.