Primes in Short Intervals

By H. Montgomery, Kannan Soundararajan. 2004

216 citations recorded by the source corpus.

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 ψ(x+H)−ψ(x), for 0≤x≤N, is approximately normal with mean ∼H and variance ∼H log N/H, when Nδ≤H≤N1−δ.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/6bf928966193e80280a7155e03265ec9e7613788

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

SIEVING AND THE ERDŐS–KAC THEOREM (2007, methodology): “Our use of this simpler technique was inspired by (Montgomery and Soundararajan, 2004).”
The variance of the number of prime polynomials in short intervals and in residue classes (2012, methodology): “A calculation based on a Hardy-Littlewood-type conjecture In the number-field setting, the problems we have considered here have previously been explored using the Hardy-Littlewood conjecture relating to the density of generalized twin primes [4, 16].”
Euler's constant: Euler's work and modern developments (2013, background): “The convergence behavior of the Buchstab function to e as x → ∞ played a crucial role in the groundbreaking 1985 work of Helmut Maier [210] showing the existence of large fluctuations in the density of primes in short intervals; this work was mentioned at the end of Section 3.”
Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem (2004, background): “Upon this we need the estimate (1.26) of Montgomery and Soundararajan for R3(h).”