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

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 · doi:10.1112/S002557930000231X
[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 · doi:10.4153/CJM-1982-095-9
[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 · doi:10.1112/S0025579300006069
[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 · eudml:204924
[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 · doi:10.3792/pja/1195526435 · minidml.mathdoc.fr
[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 · doi:10.1112/plms/s3-37.2.230
[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 · doi:10.1112/jlms/s2-10.2.153
[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 · eudml:205685