# Limitations to the equi-distribution of primes I

@article{Friedlander1989LimitationsTT, title={Limitations to the equi-distribution of primes I}, author={John B. Friedlander and Andrew Granville}, journal={Annals of Mathematics}, year={1989}, volume={129}, pages={363-382} }

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.

## 84 Citations

### Irregularities in the distribution of primes in an arithmetic progression

- Mathematics
- 1996

(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

- MathematicsScience China Mathematics
- 2019

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…

### Small gaps between primes

- MathematicsGraduate 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…

### An Asymptotic Formula for the Number of Smooth Values of a Polynomial

- Mathematics
- 1999

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…

### SOME SINGULAR SERIES AVERAGES AND THE DISTRIBUTION OF GOLDBACH NUMBERS IN SHORT INTERVALS

- Mathematics
- 1995

In this paper we shall be concerned with the question of the existence of Goldbach numbers in short intervals and the asymptotic formula for the number of representations of the even integers in a…

### On the Normal Behavior of the Iterates Of some Arithmetic Functions

- Mathematics
- 1990

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…

### Primes in short arithmetic progressions

- Mathematics
- 2015

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…

### Small gaps between products of two primes

- Mathematics
- 2009

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ⩽ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of…

### Asymptotics For Primitive Roots Producing Polynomials And Primitive Points On Elliptic Curves

- Mathematics
- 2017

Let x ≥ 1 be a large number, let f(n) ∈ Z[x] be a prime producing polynomial of degree deg(f) = m, and let u 6= ±1, v be a fixed integer. Assuming the BatemanHorn conjecture, an asymptotic counting…

### LECTURE NOTES: Sieves I

- Mathematics
- 2011

The first sieving procedure for producing primes is credited to Eratosthenes (∼200 BCE), who made a simple but important observation: if n < x and n has no prime factors < x, then n is prime. So to…

## References

SHOWING 1-10 OF 13 REFERENCES

### Limitations to the equi-distribution of primes. IV

- MathematicsProceedings 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

- Mathematics
- 1986

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

### Multiplicative Number Theory

- Mathematics
- 1967

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…

### On an asymptotic estimate of the number of numbers of an arithmetic progression which are not divisible by relatively small prime numbers (Russian)

- Mat. Sb
- 1951

### Topics in Multiplicative Number Theory The distribution of primes 15

- Lecture Notes in Mathematics
- 1971

### On the normal density of primes in small intervals and the diierence between consecutive primes

- Arch. Math. Naturvid
- 1943

### Oscillation theorems forprimes in arithmetic progressions and for sifting functions , to appear in

- J . Amer . Math . Soc .