Gaussian primes.

*(English)*Zbl 0881.11070The authors establish an asymptotic formula which shows that of the primes of the form \(l^2+m^2\) not exceeding \(x\), counted according to multiplicity of representation, about the expected number have the number \(l\) also prime. In particular this number tends to \(\infty\) with \(x\).

The treatment uses sieve ideas although, as is well known, sieve methods alone could not deliver such a result. Other ideas central to the treatment include an inequality of the large sieve type for numbers of the form \(\nu/d\) where \(\nu^2+1\equiv 0 \pmod d\), and estimates for bilinear forms over the Gaussian integers. Some of this work also provides some input into the later result of J. Friedlander and H. Iwaniec concerning prime values of \(m^2+n^4\), of which an announcement appears in [Proc. Natl. Acad. Sci. USA 94, No. 4, 1054-1058 (1997: Zbl 0870.11059)].

The authors devote some space to explaining the broader ideas of their approach as well as setting down the technical details. They also establish some easier related results in which primes are replaced by norms of ideals in abelian fields.

The treatment uses sieve ideas although, as is well known, sieve methods alone could not deliver such a result. Other ideas central to the treatment include an inequality of the large sieve type for numbers of the form \(\nu/d\) where \(\nu^2+1\equiv 0 \pmod d\), and estimates for bilinear forms over the Gaussian integers. Some of this work also provides some input into the later result of J. Friedlander and H. Iwaniec concerning prime values of \(m^2+n^4\), of which an announcement appears in [Proc. Natl. Acad. Sci. USA 94, No. 4, 1054-1058 (1997: Zbl 0870.11059)].

The authors devote some space to explaining the broader ideas of their approach as well as setting down the technical details. They also establish some easier related results in which primes are replaced by norms of ideals in abelian fields.

Reviewer: G.Greaves (Cardiff)