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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Unusually large gaps between consecutive primes

Authors: Helmut Maier and Carl Pomerance
Journal: Trans. Amer. Math. Soc. 322 (1990), 201-237
MSC: Primary 11N05; Secondary 11N35
MathSciNet review: 972703
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Abstract: Let $ G(x)$ denote the largest gap between consecutive primes below $ x$. In a series of papers from 1935 to 1963, Erdàs, Rankin, and Schànhage showed that

$\displaystyle G(x) \geq (c + o(1)){\operatorname{log}}x{\operatorname{loglog}}x{\operatorname{loglogloglog}}x{({\operatorname{logloglog}}x)^{ - 2}}$

, where $ c = {e^\gamma }$ and $ \gamma $ is Euler's constant. Here, this result is shown with $ c = {c_0}{e^\gamma }$ where $ {c_0} = 1.31256 \ldots $ is the solution of the equation $ 4/{c_0} - {e^{ - 4/{c_0}}} = 3$. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.

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Article copyright: © Copyright 1990 American Mathematical Society

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