Distribution of primes in small intervals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:21:11Z http://mathoverflow.net/feeds/question/92599 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92599/distribution-of-primes-in-small-intervals Distribution of primes in small intervals ogn 2012-03-29T19:05:59Z 2012-03-30T03:41:47Z <p>Let $\pi(x)$ be the number of primes smaller than $x$. Do there exist unconditionally universal constants $c > d$ such that $$\lim_{x \rightarrow \infty} \frac{\pi(x + \log^c x) - \pi(x)}{\log^{c-d} x} \geq 1$$</p> <p>We know that by Maier Theorem, it is not possible that $c = d+1$. </p> <p>By Selberg theorem, for any function $y(x)$ grows faster than $\log^2 x$, it holds that $$\lim_{x \rightarrow \infty} \frac{\pi(x + y) - \pi(x)}{y/\log x} = 1$$ for \emph{almost} $x$ (assuming the Riemann hypothesis). Does it hold for \emph{all} $x$ if $y(x) = \log^c x$ for some constant $c$ (with Riemann hypothesis)? </p> http://mathoverflow.net/questions/92599/distribution-of-primes-in-small-intervals/92610#92610 Answer by Alastair Irving for Distribution of primes in small intervals Alastair Irving 2012-03-29T21:01:18Z 2012-03-29T21:01:18Z <p>A weaker question is to ask for which functions $f$ the interval $[x,x+f(x)]$ contains a prime for all sufficiently large $x$. The sharpest uncoditional result is then that $f(x)\geq x^{0.525}$ is sufficient. We are therefore a long way from being able to prove results about $f(x)=\log^c x$. </p> http://mathoverflow.net/questions/92599/distribution-of-primes-in-small-intervals/92642#92642 Answer by Micah Milinovich for Distribution of primes in small intervals Micah Milinovich 2012-03-30T03:41:47Z 2012-03-30T03:41:47Z <p>Assuming the Riemann Hypothesis, I believe the best known result is due to Cramer (I cannot figure out how to add the accent of the e) and it says the following:</p> <p>There is a constant $C > 0$ such that if if the Riemann Hypothesis is true, then for every $x \ge2$ the interval $(x, x + C \sqrt{x} \log x)$ contains at least $\sqrt{x}$ prime numbers.</p> <p>This is Theorem 13.3 in Montgomery and Vaughan's <em>Multiplicative Number Theory</em>.</p> <p>Translating things from $\pi(x)$ to $\psi(x)$, exercise 2, pp. 430-431 of the same book outlines a proof that the Riemann Hypothesis implies that</p> <p>$$\psi(x+y)-\psi(x)=y+O\left(\sqrt{x} \log x \log\left(\frac{2y}{\sqrt{x} \log x}\right) \right).$$</p> <p>Thus an asymptotic holds as soon as $\frac{y}{\sqrt{x} \log x} \to \infty$. This formula simultaneously implies both Cramer's result and von Koch's well-known result that $$\psi(x) = x + O(\sqrt{x}\log^2 x) \quad \text{equivalently } \quad \pi(x) = \int_2^x \frac{dt}{\log t} + O(\sqrt{x}\log x)$$ assuming the Riemann Hypothesis.</p>