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An uncertainty principle for arithmetic sequences, preprint, available from www.arxiv.org
"... Abstract. 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 wor ..."
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Abstract. 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. 1.
Primes in almost all short intervals and the distribution of the zeros of the Riemann zetafunction
"... We study the relations between the distribution of the zeros of the Riemann zetafunction and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)#(xy) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available ..."
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We study the relations between the distribution of the zeros of the Riemann zetafunction and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)#(xy) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available zerofree regions for the Riemann zetafunction, and also on the strength of density bounds for the zeros themselves. We also study implications in the opposite direction: assuming that an asymptotic formula like the above is valid for almost all x in a given range of values for y, we find zerofree regions or density bounds.
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 ..."
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: 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 !...
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 why do natural logs a ..."
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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