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Published online by Cambridge University Press: 10 September 2021
This chapter gives a sample of work done on prime gaps spanning more than half of the twentieth century, showing the importance of the problem and giving a context for what was to follow. Erdős proved in 1940 that there are an infinite number of consecutive primes strictly less than the average gap. His proof is given in Section 3.3. Section 3.4 gives the theorem of Bombieri and Davenport showing that there are an infinite number of primes less than half the average gap. Their proof uses many advanced methods and tools with a fundamental lemma of 10 steps. Section 3.5 gives Granville and Soundararajan’s version of the work of Maier, which has applications to primes in intervals as well as prime gaps.
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