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PRIME NUMBERS IN INTERVALS STARTING AT A FIXED POWER OF THE INTEGERS

Published online by Cambridge University Press:  01 August 2009

DANILO BAZZANELLA
DANILO BAZZANELLA*
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy (email: [email protected])
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Abstract

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The best known results about the distribution of prime numbers in short intervals imply that all intervals [n,n+H]⊂[N,2N] contain the expected number of primes for all H≥N7/12, and almost all intervals [n,n+H]⊂[N,2N] contain the expected number of primes for all H≥N1/6. As a natural generalization, this paper is concerned with the distribution of prime numbers in intervals of type [nα,nα+H] with α>1.

Keywords

prime numbers in short intervals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 1 , August 2009 , pp. 83 - 99
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Bazzanella, D., ‘Primes between consecutive square’, Arch. Math. 75 (2000), 29–34.CrossRefGoogle Scholar
[2]Bazzanella, D. and Perelli, A., ‘The exceptional set for the number of primes in short intervals’, J. Number Theory 80 (2000), 109–124.CrossRefGoogle Scholar
[3]Davenport, H., Multiplicative Number Theory, 2nd edn, Graduate Texts in Mathematics, 74 (Springer, Berlin, 1980).CrossRefGoogle Scholar
[4]Graham, S. W. and Kolesnik, G., Van der Corput’s Method of Exponential Sums (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[5]Heath-Brown, D. R., ‘The differences between consecutive primes II’, J. London Math. Soc. (2) 19 (1979), 207–220.CrossRefGoogle Scholar
[6]Heath-Brown, D. R., ‘Zero density estimates for the Riemann zeta-function and Dirichlet L-function’, J. London Math. Soc. (2) 19 (1979), 221–232.CrossRefGoogle Scholar
[7]Heath-Brown, D. R., ‘The number of primes in a short interval’, J. Reine Angew. Math. 389 (1988), 22–63.Google Scholar
[8]Huxley, M. N., ‘On the difference between consecutive primes’, Invent. Math. 15 (1972), 164–170.CrossRefGoogle Scholar
[9]Ivić, A., The Riemann Zeta-Function (John Wiley and Sons, New York, 1985).Google Scholar
[10]Montgomery, H. L. and Vaughan, R .C., ‘The large sieve’, Mathematika 20 (1973), 119–134.CrossRefGoogle Scholar
[11]Selberg, A., ‘On the normal density of primes in small intervals, and the difference between consecutive primes’, Arc. Math. Naturvid. 47(6) (1943), 87–105.Google Scholar
[12]Titchmarsh, E. C., The Theory of the Riemann Zeta-function, 2nd edn (Oxford University Press, Oxford, 1986).Google Scholar
[13]Yu, G., ‘The differences between consecutive primes’, Bull. London Math. Soc. 28(3) (1996), 242–248.CrossRefGoogle Scholar
[14]Zaccagnini, A., ‘Primes in almost all short intervals’, Acta Arith. 84 (1998), 225–244.CrossRefGoogle Scholar