Primes in short intervals

Abstract

Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of $\psi (x+H)-\psi (x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^{\delta }\le H\le N^{1-\delta }$.

29 pages

Cited by in corpus (12)

• Unexpected biases in the distribution of consecutive primes
• Hidden Multiscale Order in the Primes
• On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class
• The jumping champion conjecture
• Pair correlation and twin primes revisited
• Lemke Oliver and Soundararajan bias for consecutive sums of two squares
• The covariance of almost-primes in ${\mathbb{F}}_q[T]$
• Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in ${\mathbb{F}}_q(t)$
• The distribution of $k$-tuples of reduced residues
• The variance of integers without small prime factors in short intervals
• Squarefrees are Gaussian in short intervals
• Twin prime correlations from the pair correlation of Riemann zeros