Perelli, A.; Pintz, J.; Salerno, S. Bombieri’s theorem in short intervals. (English) Zbl 0579.10019 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 Cited in 2 ReviewsCited in 11 Documents 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 PDF BibTeX XML Cite \textit{A. Perelli} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, 529--539 (1984; Zbl 0579.10019) Full Text: Numdam EuDML OpenURL References: [1] E. Bombieri , Le Grand Crible dans la Théorie Analytique des Nombres , Astérisque , no. 18 ( 1974 ). MR 891718 | Zbl 0292.10035 · Zbl 0292.10035 [2] H. Davenport , Multiplicative Number Theory, II edition , Springer-Verlag , 1980 . MR 606931 | Zbl 0453.10002 · Zbl 0453.10002 [3] P.X. Gallagher , Bombieri’s mean value theorem , Mathematika , 15 ( 1968 ), pp. 1 - 6 . MR 237442 | Zbl 0174.08103 · Zbl 0174.08103 [4] D.R. Heath-Brown , Prime numbers in short intervals and a generalized Vaughan identity , Canad. J. Math. , 34 ( 1982 ), pp. 1365 - 1377 . MR 678676 | Zbl 0478.10024 · Zbl 0478.10024 [5] D.R. Heath-Brown , Sieve identities and gaps between primes , Astérisque , no. 94 ( 1982 ), pp. 61 - 65 . Zbl 0502.10029 · Zbl 0502.10029 [6] M.N. Huxley - H. Iwaniec , Bombieri’s theorem in short intervals , Mathematika , 22 ( 1975 ), pp. 188 - 194 . MR 389790 | Zbl 0317.10048 · Zbl 0317.10048 [7] M. Jutila , A. statistical density theorem for L-functions with applications , Acta Arith. , 16 ( 1969 ), pp. 207 - 216 . Article | MR 252336 | Zbl 0185.10901 · Zbl 0185.10901 [8] A.F. Lavrik , An approximate functional equation for the Dirichlet L-functions , Trans. Moscow Math. Soc. , 18 ( 1968 ), pp. 101 - 115 . MR 236126 | Zbl 0195.33301 · Zbl 0195.33301 [9] H.L. Montgomery , Topics in Multiplicative Number Theory , Springer L.N. no. 227 ( 1971 ). MR 337847 | Zbl 0216.03501 · Zbl 0216.03501 [10] Y. Motohashi , On a mean value theorem for the remainder term in the prime number theorem for short arithmetical progressions , Proc. Japan Acad. Ser A Math. Sci. , 47 ( 1971 ), pp. 653 - 657 . Article | MR 304329 | Zbl 0246.10027 · Zbl 0246.10027 [11] K. Prachar , Primzahtverteitung , Springer-Verlag ( 1957 ). MR 87685 | Zbl 0080.25901 · Zbl 0080.25901 [12] S.J. Ricci , Mean-values theorems for primes in short intervals , Proc. London Math. Soc. , ( 3 ) 37 ( 1978 ), pp. 230 - 242 . MR 507605 | Zbl 0399.10043 · Zbl 0399.10043 [13] R.C. Vaughan , Mean value theorems in prime number theory , J. London Math. Soc. , ( 2 ) 10 ( 1975 ), pp. 153 - 162 . MR 376567 | Zbl 0314.10028 · Zbl 0314.10028 [14] R.C. Vaughan , An elementary method in prime number theory , Acta Arith. , 37 ( 1980 ), pp. 111 - 115 . Article | MR 598869 | Zbl 0448.10037 · Zbl 0448.10037 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.