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A generalization of Bombieri’s prime number theorem to algebraic number fields. (English) Zbl 0605.10023
Bombieri’s mean value theorem yields (in the usual notation) \[ \sum_{q\leq Q}\max_{y\leq x}\max_{(\ell,q)=1}| \pi (y;q,\ell) - \frac{li(y)}{\phi (q)}| \ll x(\log x)^{-A} \] for any \(A>0\) with \(Q=x^{1/2} (\log x)^{-B}\), where \(B\) and the implied constant depend on \(A\) alone.
The intent of the present paper is to generalize this central result in prime number theory to algebraic number fields \(K\) of finite degree over the rationals. The primes \(p\) counted in \(\pi (y;q,\ell)\) are replaced by totally positive algebraic prime numbers \(\omega\in K\) lying in certain parallelotopes and satisfying \(\omega\equiv \alpha \bmod {\mathfrak q}\), where \({\mathfrak q}\) is an integral ideal of \(K\). An integer \(\omega\in K\) is said to be a prime number in \(K\), if the principal ideal (\(\omega)\) is a prime ideal. The method of proof is influenced by R. C. Vaughan’s work [ibid. 37, 111–115 (1980; Zbl 0448.10037)] on the corresponding result in the rational case.
The principle underlying the treatment of Bombieri’s theorem is that of the large sieve. Here, the basic inequality of the large sieve method in \(K\) is required in a new form given by the author in [Manuscr. Math. 57, 181–194 (1987; Zbl 0594.10038)]. Moreover, the author’s version of the Pólya-Vinogradov character sum estimate in the setting of a number field [J. Number Theory 17, 52–70 (1983; Zbl 0511.10028)] plays a fundamental role in the proof. The given generalization of Bombieri’s result can be used in many of the most significant applications of sieve theory in \(K\).

11N35 Sieves
11N13 Primes in congruence classes
11R47 Other analytic theory
11L40 Estimates on character sums
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