The Cram\'er conjecture holds with a positive probability
classification
🧮 math.NT
keywords
positivecontainfixedmultiplenumberprimeproportionscalar
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We prove that a positive proportion of the intervals of any fixed scalar multiple of $\log(X)$ in the dyadic interval $[X,2X]$ contain a prime number. We also show that a positive proportion of the congruence classes modulo $q$ contain a prime number smaller than any fixed scalar multiple of $\varphi(q)\log(q).$
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