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On the Riemann hypothesis and the difference between primes

Adrian W. Dudek

Mathematical Sciences Institute, The Australian National University, Canberra, 0200, Australia

https://doi.org/10.1142/S1793042115500426Cited by:13 (Source: Crossref)

Abstract

We prove some results concerning the distribution of primes assuming the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval for all x ≥ 2; this improves a result of Ramaré and Saouter. We then show that the constant 4/π may be reduced to (1 + ϵ) provided that x is taken to be sufficiently large. From this, we get an immediate estimate for a well-known theorem of Cramér, in that we show the number of primes in the interval is greater than for c = 3 + ϵ and all sufficiently large values of x.

Keywords:

AMSC: 11N05, 11M26, 11M36

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