Bombieri’s theorem in short intervals. (English) Zbl 0579.10019

Perelli, A.; Pintz, J.; Salerno, S.

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, 529-539 (1984).

Using D. R. Heath-Brown’s extension of Vaughan’s identity [Can. J. Math. 34, 1365-1377 (1982; Zbl 0478.10024)] the authors prove the following short-interval-version of Bombieri’s mean value theorem \[ \sum_{q\leq Q}\max_{(a,q)=1}\max_{h\leq y}\max_{x/2<z\leq x}\quad | \Psi (z+h,q;a)-\Psi (z,q;a)-h/\phi (q)| \ll y/(\log x)^ A \] where \(y=x^{\vartheta}\), \(Q=x^{\vartheta '}/(\log x)^ B\), \(\vartheta >3/5\), \(\vartheta\) ’\(\leq \vartheta -1/2\), A and B positive. {See also the following review of Part II.}

Reviewer: D.Wolke

MSC:

11N13	Primes in congruence classes
11N05	Distribution of primes

Keywords:

short intervals; Heath-Brown’s identity; Bombieri’s mean value theorem

Citations:

Zbl 0579.10020; Zbl 0478.10024

References:

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