ON THE DISTRIBUTION OF M-TUPLES OF B-NUMBERS

Werner Georg Nowak

Abstract: 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 $\Qi(i)$.

Keywords: $B$-numbers; Selberg sieve; norms of ideals in number fields

Classification (MSC2000): 11P05, 11N35

Full text of the article: (for faster download, first choose a mirror)

• DVI file (32 kilobytes)
• Compressed PostScript file (72 kilobytes)
• PDF file (208 kilobytes)