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Primes in short intervals
 Commun. Math. Phys
"... Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday. Abstract. Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approxima ..."
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Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday. Abstract. Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log N/H, when N δ ≤ H ≤ N 1−δ. Cramér [4] modeled the distribution of prime numbers by independent random variables Xn (for n ≥ 3) that take the value 1 (n is “prime”) with probability 1 / logn and take the value 0 (n is “composite”) with probability 1 − 1 / log n. If pn denotes the n th prime
THE DISTRIBUTION OF PRIME NUMBERS
, 2006
"... What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consec ..."
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What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consecutive primes The prime number theorem tells us that π(x), the number of primes below x, is ∼ x / logx. Equivalently, if pn denotes the nth smallest prime number then pn ∼ n log n. What is the distribution of the gaps between consecutive primes, pn+1 − pn? We have just seen that pn+1 − pn is approximately log n “on average”. How often do we get a gap of size 2 logn, say; or of size 1 log n? One way to make this question precise 2 is to fix an interval [α, β] (with 0 ≤ α < β) and ask for
Different Approaches to the Distribution of Primes
 MILAN JOURNAL OF MATHEMATICS
, 2009
"... In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction. ..."
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In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction.
The influence of the first term of an arithmetic progression
 IN PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2011
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An uncertainty principle for function fields
, 2008
"... In a recent paper, Granville and Soundararajan [8] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be welldistributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of [8] and prove tha ..."
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In a recent paper, Granville and Soundararajan [8] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be welldistributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of [8] and prove that a similar phenomenon holds in Fq[t].
THE ELLIOTTHALBERSTAM CONJECTURE IMPLIES THE VINOGRADOV LEAST QUADRATIC NONRESIDUE CONJECTURE
"... Abstract. For each prime p, let nppq denote the least quadratic nonresidue modulo p. Vinogradov conjectured that nppq “ Oppεq for every fixed ε ą 0. This conjecture follows from the generalised Riemann hypothesis, but remains open in general. In this paper we show that Vinogradov’s conjecture follow ..."
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Abstract. For each prime p, let nppq denote the least quadratic nonresidue modulo p. Vinogradov conjectured that nppq “ Oppεq for every fixed ε ą 0. This conjecture follows from the generalised Riemann hypothesis, but remains open in general. In this paper we show that Vinogradov’s conjecture follows from the ElliottHalberstam conjecture on the distribution of primes in arithmetic progressions. We also give a variant of this argument that obtains bounds on short centred character sums from “Type II ” estimates of the type introduced recently by Zhang and improved upon by the Polymath project. In particular, we can obtain an improvement over the Burgess bound would be obtained if one had Type II estimates with level of distribution above 2{3 (when the conductor is not cubefree) or 3{4 (if the conductor is cubefree). Some applications to the least primitive root are also given. 1.
BUBBLES OF CONGRUENT PRIMES
"... Abstract. In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu’s theorem to imaginary quadratic fields, where we prove the existence of “bubbles ” containing arbi ..."
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Abstract. In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu’s theorem to imaginary quadratic fields, where we prove the existence of “bubbles ” containing arbitrarily many primes which are all, up to units, congruent to a modulo q. 1.
MAIER MATRICES BEYOND Z
"... This paper is an expanded version of a talk given at the 2007 Integers Conference, giving an overview of the Maier matrix method and surveying the author’s work in extending it beyond the integers. 1. Maier Matrices Loosely speaking, a Maier matrix is a combinatorial device used to prove the existen ..."
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This paper is an expanded version of a talk given at the 2007 Integers Conference, giving an overview of the Maier matrix method and surveying the author’s work in extending it beyond the integers. 1. Maier Matrices Loosely speaking, a Maier matrix is a combinatorial device used to prove the existence of irregular or interesting patterns in the distribution of primes or related sequences. We will illustrate the technique with two particularly interesting examples. The first is Maier’s 1985 proof [6] that “unexpected ” irregularities exist in the distribution of primes in short intervals. In particular, Maier proved that for any A> 0 there exists a constant δA> 0 such that lim sup n→∞ π(n + log A n) − π(n) log A−1 n ≥ 1 + δA, lim inf n→∞ π(n + log A n) − π(n) log A−1 n ≤ 1 − δA. (0.1) These irregularities are unexpected in the sense that they contradict probabilistic heuristics for A> 2. The proof is as follows. For a variable y, let Q = ∏ p<y p, let x1 = QD for some fixed large D, and let C be a parameter to be determined later. Consider the following matrix of integers: ⎡