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Bombieri's theorem in short intervals

Published online by Cambridge University Press:  26 February 2010

M. N. Huxley and
H. Iwaniec
M. N. Huxley
Affiliation:
Department of Pure Mathematics, University College, Cardif
H. Iwaniec
Affiliation:
Mathematics Institute, Polish Academy of Sciences, Warsaw, Poland
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Extract

Let x, y, Q denote large real numbers, with x > y > Q. The object of this paper is to prove the following result

for arbitrary A > 0, with a larger value of Q than hitherto. The symbol denotes as usual the suppression of an absolute constant, π(N; q, a) denotes the number of prime numbers up to N which are congruent to a (mod q), and ϕ(q) denotes Euler's function.

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Type
Research Article
Information
Mathematika , Volume 22 , Issue 2 , December 1975, pp. 188 - 194
Copyright
Copyright © University College London 1975

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References

1.Bombieri, E.. “On the large sieve”, Mathematika, 12 (1965), 201–225.CrossRefGoogle Scholar
2.Fogels, E.. “On average values of arithmetic functions”, Proc. Camb. Phil. Soc, 37 (1941), 358–372.CrossRefGoogle Scholar
3.Gallagher, P. X.. “Bombieri's mean value theorem”, Mathematika, 15 (1968), 1–6.CrossRefGoogle Scholar
4.Huxley, M. N.. “On the difference between consecutive primes”, Irwentiones Math., 15 (1972), 164–170.CrossRefGoogle Scholar
5.Huxley, M. N.. “Large values of Dirichlet polynomials III”, Acta Arithmetica 26 (1974), 431–140.Google Scholar
6.Iwaniec., H. “The half dimensional sieve” Acta Arithmetica (to appear).Google Scholar
7.Jutila, M.. “A statistical density theorem for p.-functions with applications”, Acta Arithmetica, 16 (1969), 207–216.CrossRefGoogle Scholar
8.Kolesnik, G. A.. “On the estimation of some trigonometric sums”, Acta Arithmetica, 25 (1973) 7–30.Google Scholar
9.Montgomery, H. L.. Topics in multiplicative number theory, Lecture notes in Mathematics, 227 (Springer, 1971).CrossRefGoogle Scholar
10.Motohashi, Y.. “On a mean value theorem for the remainder term in the prime number theorem for short arithmetic progressions”, Proc. Japan. Acad., 47 (1971), 653–657.CrossRefGoogle Scholar
11.Prachar, K.. Primzahlverteilung, (Springer 1957).Google Scholar
12.Vaughan, R. C.. “Mean value theorems in prime number theory”, J. London Math. Soc. (2), 10 (1975), 153–162.CrossRefGoogle Scholar
13.Vinogradov, A. I.. “On the density hypothesis for Dirichlet L-functions”, Izv. Akad. Nauk SSSR Ser. Mat., 29 (1965), 903–934; correction, Izv. Akad. Nauk SSSR Ser. Mat., 30 (1966), 719–720.Google Scholar