A note on primes in short intervals
01 Mar 2006-International Journal of Number Theory (World Scientific Publishing Company)-Vol. 02, Iss: 01, pp 105-110
TL;DR: In this paper, it was shown that an average form of the Hardy-Littlewood conjecture suffices for the Gaussian distribution of primes in short intervals assuming a quantitative Hardy-littlewood conjecture.
Abstract: Montgomery and Soundararajan obtained evidence for the Gaussian distribution of primes in short intervals assuming a quantitative Hardy–Littlewood conjecture. In this article, we show that their methods may be modified and an average form of the Hardy–Littlewood conjecture suffices.
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TL;DR: In this paper, an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution is given, including a discussion of the relation between equilibria.
Abstract: This is an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution.
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TL;DR: In this paper, it was shown that the lower-order terms in the size of the Gaussian moment constants of the singular series can be computed in the function field setting when the number of moments is odd.
Abstract: Montgomery and Soundararajan showed 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 $N^{\delta} \le H \le N^{1-\delta}$. Their work depends on showing that sums $R_k(h)$ of $k$-term singular series are $\mu_k(-h \log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k) + \varepsilon})$, where $A$ is a constant and $\mu_k$ are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when $k$ is odd, $R_k(h) \asymp h^{(k-1)/2}(\log h)^{(k+1)/2}$. We prove an upper bound with the correct power of $h$ when $k = 3$, and prove analogous upper bounds in the function field setting when $k =3$ and $k = 5$. We provide further evidence for this conjecture in the form of numerical computations.
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TL;DR: In this article, it was shown 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 N δ ≤ H ≤ N 1−δ.
Abstract: Contrary to what would be predicted on the basis of Cramer's model con- cerning 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 N δ ≤ H ≤ N 1−δ .
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