Limitations to the equi-distribution of primes I

  title={Limitations to the equi-distribution of primes I},
  author={John B. Friedlander and Andrew Granville},
  journal={Annals of Mathematics},
In an earlier paper FG] we showed that the expected asymptotic formula (x; q; a) (x)==(q) does not hold uniformly in the range q < x= log N x, for any xed N > 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept xed. However, by a new construction, we show herein that this fails in the same ranges, for a xed and, indeed, for almost all a satisfying 0 < jaj < x= log N x. 

Irregularities in the distribution of primes in an arithmetic progression

(1) |∆(x; q, a)| e (q/x)1/2−e log x uniformly for q ≤ x, for any given e > 0. Recently, Friedlander and Granville [1] disproved Montgomery’s conjecture (1). They showed that for any A > 0 there exist

Primes in arithmetic progressions with friable indices

We consider the number π(x, y; q, a) of primes p ⩽ x such that p ≡ a (mod q ) and ( p − a )/ q is free of prime factors greater than y. Assuming a suitable form of Elliott-Halberstam conjecture, it

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  • A. Mocanu
  • Mathematics
    Graduate Studies in Mathematics
  • 2019
In this paper we study the work of James Maynard [10], in which he proves that lim inf n→∞ (pn+m − pn) <∞, thus establishing that there are infinitely many intervals of finite length that contain a

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Although we expect to find many smooth numbers (i.e., numbers with no large prime factors) among the values taken by a polynomial with integer coefficients, it is unclear what the asymptotic number


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On the Normal Behavior of the Iterates Of some Arithmetic Functions

Let ϕ1(n) = ϕ(n) where ϕ is Euler’s function, let ϕ2(n) = ϕ(ϕ(n)), etc. We prove several theorems about the normal order of ϕk(n) and state some open problems. In particular, we show that the normal

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Let x, h and Q be three parameters. We show that, for most moduli q ≤ Q and for most positive real numbers y ≤ x, every reduced arithmetic progression a(mod q) has approximately the expected number

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Limitations to the equi-distribution of primes. IV

  • J. FriedlanderA. Granville
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1991
We construct infinitely many different polynomials of given degree which take either significantly more or significantly less prime values than expected.

Primes in arithmetic progressions to large moduli

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Multiplicative Number Theory

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The

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Topics in Multiplicative Number Theory The distribution of primes 15

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