Results 1 
8 of
8
An uncertainty principle for arithmetic sequences
, 2004
"... Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.
Primes in almost all short intervals and the distribution of the zeros of the Riemann zetafunction
"... ..."
Limitations to the Equidistribution of Primes III
 Comp. Math
, 1992
"... : In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept f ..."
Abstract
 Add to MetaCart
: In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept fixed. However, by a new construction, we show herein that this fails in the same ranges, for a fixed and, indeed, for almost all a satisfying 0 ! jaj ! x= log N x. 1. Introduction. For any positive integer q and integer a coprime to q, we have the asymptotic formula (1:1) ß(x; q; a) ¸ ß(x) OE(q) as x ! 1, for the number ß(x; q; a) of primes p x with p j a (mod q), where ß(x) is the number of primes x, and OE is Euler's function. In fact (1.1) is known to hold uniformly for (1:2) q ! log N x and all (a; q) = 1, for every fixed N ? 0 (the SiegelWalfisz Theorem), for almost all q ! x 1=2 = log 2+" x and all (a; q) = 1 (the BombieriVinogradov Theorem) and for almost all q !...
Variants of the Selberg sieve, and bounded . . .
, 2014
"... For any m ≥ 1, let Hm denote the quantity lim infn→∞(pn+m − pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H1 ≤ 4680, and then by Maynard to H1 ≤ 600, who also established for ..."
Abstract
 Add to MetaCart
For any m ≥ 1, let Hm denote the quantity lim infn→∞(pn+m − pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H1 ≤ 4680, and then by Maynard to H1 ≤ 600, who also established for the first time a finiteness result for Hm form ≥ 2, and specifically that Hm m3e4m. If one also assumes the ElliottHalberstam conjecture, Maynard obtained the bound H1 ≤ 12, improving upon the previous bound H1 ≤ 16 of Goldston, Pintz, and Yıldırım, as well as the bound Hm m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1 ≤ 246 unconditionally and H1 ≤ 6 under the assumption of the generalized ElliottHalberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1, h2, h3), there are infinitely many n for which at least two of n + h1, n + h2, n + h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem ’ argument of Selberg to show that the H1 ≤ 6 bound is the best possible that one can obtain from purely sievetheoretic considerations. For largerm, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound Hm me
A Heuristic for the Prime Number Theorem This article appeared in The Mathematical Intelligencer 28:3 (2006) 6–9, and is copyright by Springer
"... Why does ‰ play such a central role in the distribution of prime numbers? Simply citing the Prime Number Theorem (PNT), which asserts that pHxL ~ x ê ln x, is not very illuminating. Here "~ " means "is asymptotic to " and pHxL is the number of primes less than or equal to x. So w ..."
Abstract
 Add to MetaCart
Why does ‰ play such a central role in the distribution of prime numbers? Simply citing the Prime Number Theorem (PNT), which asserts that pHxL ~ x ê ln x, is not very illuminating. Here "~ " means "is asymptotic to " and pHxL is the number of primes less than or equal to x. So why do natural logs appear, as opposed to another flavor of logarithm? The problem with an attempt at a heuristic explanation is that the sieve of Eratosthenes does not behave as one might guess from pure probabilistic considerations. One might think that sieving out the composites under x using primes up to è!!! x would lead to x P è!!! J1p< x 1 ÅÅÅÅ N as an asymptotic estimate of the count of p numbers remaining (the primes up to x; p always represents a prime). But this quantity turns out to be not asymptotic to x ê ln x. For F. Mertens proved in 1874 that the product is actually asymptotic to 2 ‰g ê ln x, or about 1.12 ê ln x. Thus the sieve is 11 % (from 1 ê 1.12) more efficient at eliminating composites than one might expect. Commenting on this phenomenon, which one might call the Mertens Paradox, Hardy and
A Survey of Results on Primes in Short Intervals
"... Prime numbers have been a source of fascination for mathematicians since antiquity. The proof that there are infinitely many prime numbers is attributed to Euclid (fourth century B.C.). The basic method of determining all primes less than a given number N is ..."
Abstract
 Add to MetaCart
(Show Context)
Prime numbers have been a source of fascination for mathematicians since antiquity. The proof that there are infinitely many prime numbers is attributed to Euclid (fourth century B.C.). The basic method of determining all primes less than a given number N is