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On large differences between consecutive primes
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abstract
We show that $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+\epsilon}$$ and $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+\epsilon},$$ where $p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent work with Harman's sieve, improving and extending his results. We give applications of the results to prime-representing functions, binary digits of primes and approximation of reals by multiplicative functions.
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Cited by 1 Pith paper
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On the number of exceptional intervals to the prime number theorem in short intervals
An explicit formula expresses the exceptional-set exponent for the short-interval prime number theorem in terms of zero density estimates, yielding improved numerical bounds such as μ(17/30) ≤ 7/12.
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