No arithmetic sequence is very well-distributed

Number theory 1 Minute

You need to know: Prime numbers, logarithm, O(1) notation,

Background: Let {\cal P} be the set of primes. For x>0, let {\cal P}(x)=\{p\in {\cal P}: p<x\} be the set of primes less than x, \theta(x)=\sum\limits_{p\in{\cal P}(x)} \log p, and let \Delta(x,y) = \frac{\theta(x+y) - \theta(x) - y}{y}.

The Theorem: On 1st June 2004, Andrew Granville and Kannan Soundararajan submitted to arxiv and the Annals of Mathematics a paper in which they proved, among other results, the following theorem. Let x be large and y be such that \log x \leq y \leq \exp\left(\frac{\beta \sqrt{\log x}}{2\sqrt{\log\log x}}\right), where \beta>0 is an absolute constant. Then there exist numbers x_+ and x_- in (x,2x) such that \Delta(x_+,y) \geq y^{-\delta(x,y)} and \Delta(x_-,y) \leq -y^{-\delta(x,y)}, where \delta(x,y) = \frac{1}{\log\log x}\left( \log\left(\frac{\log y}{\log \log x}\right) + \log\log \left(\frac{\log y}{\log \log x}\right) + O(1) \right).

Short context: Function \theta(x) counts primes up to x, each prime p with weight \log p. Famous prime number theorem states that that \theta(x) \approx x for large x, hence the number of weighted primes on interval (x, x+y] is about \theta(x+y) - \theta(x) \approx y. Function \Delta(x,y) measures the quality of this approximation, and the Theorem states that there are intervals with much more and much less primes than average. In author’s words, primes are not very well distributed. In fact, the authors proved a much more general (and surprising) result that the same holds for any “arithmetic sequence”, but the exact definition of this is too difficult to be presented here.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.3 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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