Distribution of Primes in Short Intervals

Understanding how primes are distributed at fine scales -- in short intervals $[x, x+y]$ -- requires going beyond the prime number theorem. The key tools are zero-density estimates for $\zeta(s)$ and explicit formulas.

Prime Gaps

Definition6.1Prime Gaps and Cramér's Conjecture

The $n$ -th prime gap is $g_n = p_{n+1} - p_n$ . The PNT implies $g_n = o(p_n)$ , and unconditionally $g_n \ll p_n^{0.525}$ (Baker-Harman-Pintz, 2001). Cramér's conjecture asserts $g_n = O((\log p_n)^2)$ . Under RH: $g_n \ll p_n^{1/2}\log p_n$ . Maynard and Ford-Green-Konyagin-Tao proved that large gaps $g_n \gg \log p_n \cdot \frac{\log\log p_n \cdot \log\log\log\log p_n}{\log\log\log p_n}$ occur infinitely often.

Definition6.2Primes in Short Intervals

The interval $[x, x+y]$ contains $\sim y/\log x$ primes if $y/x \to \infty$ slowly enough. Huxley (1972) proved $\pi(x+y) - \pi(x) \sim y/\log x$ for $y \geq x^{7/12+\varepsilon}$ . Under RH, this holds for $y \geq x^{1/2+\varepsilon}$ . The short interval PNT $\psi(x+y) - \psi(x) \sim y$ depends on controlling the contribution of zeta zeros near $\sigma = 1$ .

Zero-Density Estimates

ExampleZero-Density Theorems

Let $N(\sigma, T) = |\{\rho = \beta + i\gamma : L(\rho, \chi) = 0, \beta \geq \sigma, |\gamma| \leq T\}|$ . Ingham's bound: $N(\sigma, T) \ll T^{3(1-\sigma)/(2-\sigma)+\varepsilon}$ . The density hypothesis asserts $N(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ . Zero-density estimates substitute for RH in many applications: they show that zeros near $\sigma = 1$ are rare enough that their collective contribution is manageable.

RemarkExplicit Formula and Error Terms

The explicit formula $\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} + O(\log^2 x)$ shows that the error in the PNT is controlled by the sum over zeros. The zero-free region $\sigma > 1 - c/\log t$ implies each zero contributes $|x^\rho/\rho| \leq x^{1-c/\log T}/|\rho|$ . Summing over zeros with $|\gamma| \leq T$ and optimizing $T$ gives $|\psi(x) - x| \ll x\exp(-c\sqrt{\log x})$ , the de la Vallee-Poussin error term.