Primes in short intervals.

An uncertainty principle for arithmetic sequences

“…Our discussions are motivated by a general result of K. F. Roth [15] on irregularities of distribution, and a particular result of H. Maier [11] which imposes restrictions on the equidistribution of primes.…”

“…We are more interested here in sets of integers that arise in arithmetic, such as the primes. In [11] H. Maier developed an ingenious method to show that for any A ≥ 1 there are arbitrarily large x such that the interval (x, x + (log x) A ) contains significantly more primes than usual (that is, ≥ (1 + δ A )(log x) A−1 primes for some δ A > 0) and also intervals (x, x + (log x) A ) containing significantly fewer primes than usual. Adapting his method J. Friedlander and A. Granville [3] showed that there are arithmetic progressions containing significantly more (and others with significantly fewer) primes than usual.…”

Different Approaches to the Distribution of Primes

“…However this is not true. In 1984, Maier [50] gave a delightful sieve theory argument to show that for any constant A > 2 there exists a constant δ A > 0 such that there are arbitrarily large integers x and X for which…”

Iterated Absolute Values of Differences of Consecutive Primes

“…(There is a conjecture of Cramer [5] that these gaps are 0((logx)2), and numerical evidence [3,4,20] supports this conjecture as well as a slightly stronger one of Shanks [16]. There are heuristic arguments, based on work of Maier [9], that suggest the true order of magnitude might be slightly larger, but at most by some fractional power of log log x.) However, the best published result is that of Mozzochi [10], namely that these gaps are < x0-548 for large x , and even on the assumption of the Riemann Hypothesis the bound for gaps can currently be lowered only to Table 2.…”