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Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

Authors:	H. Kadiri.

Affiliation:	Département de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada

Abstract:	For any and any non-exceptional modulus , we prove that, for large enough ( $xge alpha_{epsilon}log^2 q$ ), the interval $left[ e^x,e^{x+epsilon }right]$ contains a prime in any of the arithmetic progressions modulo . We apply this result to establish that every integer larger than is a sum of seven cubes.

Keywords:	Analytic number theory Dirichlet $L$-functions primes sums of cubes

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