## Prime Number Races

Venue: | Amer. Math. Monthly |

Citations: | 13 - 1 self |

### BibTeX

@ARTICLE{Granville_primenumber,

author = {Andrew Granville and Greg Martin},

title = {Prime Number Races},

journal = {Amer. Math. Monthly},

year = {},

pages = {1--33}

}

### OpenURL

### Abstract

1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking

### Citations

257 |
A Course in Arithmetic
- Serre
- 1973
(Show Context)
Citation Context ...ke ∞∏ q (1 − q n )(1 − q 23n ∞∑ ) = anq n n=1 appear frequently in arithmetic geometry: this is an example of a modular form, and it has all sorts of seemingly miraculous properties (see Serre’s book =-=[23]-=-). One such miracle is that for every prime p the value of ap is 2, 0, or −1 and that the proportions of each are around 1/6, 1/2, and 1/3, respectively. Under the appropriate assumptions Ng showed th... |

140 |
Some problems of ‘partitio numerorum’; iii: On the expression of a number as a sum of primes
- Hardy, Littlewood
- 1923
(Show Context)
Citation Context ... 2k foranyvalueof2k whatsoever. Nonetheless, we can predict how many there should be. The following conjecture as to how many such pairs there are up to x is due, essentially, to Hardy and Littlewood =-=[12]-=-: The Hardy-Littlewood Conjecture. Let k be a positive integer, and let π2k(x) be the number of prime pairs (p, p + 2k) with p ≤ x. Then π2k(x) ∼ 2C2 ∏ p|k,p>2 ( ) p − 1 · Li2(x), p − 2 where C2 = ∏ p... |

119 |
Multiplicative Number Theory
- Davenport
- 1980
(Show Context)
Citation Context ... Their compelling argument runs as follows: Consider the (k − 1)-dimensional vector whose ith entry is the difference January 2006] PRIME NUMBER RACES 25#{primes qn + ai ≤ x}−#{primes qn + ai+1 ≤ x} =-=(5)-=- for i = 1, 2,...,k − 1. Notice that the k counting functions #{primes qn + ai ≤ x} are all tied with one another precisely if this (k − 1)-dimensional vector has the value (0, 0,...,0). Asweletx incr... |

59 |
Titchmarsh, The Theory of the Riemann Zeta-Function
- C
- 1986
(Show Context)
Citation Context ...ex numbers. In this case, the definition of ζ(s) as presented works only when σ>1, but it turns out that analytic continuation allows us to define ζ(s) for every complex number s other than s = 1(see =-=[26]-=- for details). This description of the process of analytic continuation looks disconcertingly magical. Fortunately, there is a quite explicit way to show how ζ(σ + it) can be “sensibly” defined at lea... |

36 |
Distribution des nombres premiers
- Littlewood
- 1914
(Show Context)
Citation Context ...ong stretch. Nonetheless, given this data, one might guess that 4n + 1 will occasionally take the lead as we continue to watch this marathon. Indeed this is the case, as Littlewood discovered in 1914 =-=[18]-=-: Theorem (J.E. Littlewood, 1914). There are arbitrarily large values of x for which there are more primes of the form 4n + 1 up to x than primes of the form 4n + 3. In fact, there are arbitrarily lar... |

9 | Biases in the Shanks-Rényi prime number race, Experim - Feuerverger, Martin |

6 |
Quadratic residues and the distribution of primes
- Shanks
- 1959
(Show Context)
Citation Context ...iple of ln ln ln x for infinitely many values of x. 10 The infinite extent of the histogram is certainly not evident from figure 5, but this is not so surprising since, as Dan Shanks remarked in 1959 =-=[24]-=-: ln ln ln x goes to infinity with great dignity. In (3) we saw how the difference between π(x) and Li(x) can be approximated by a sum of waves whose frequencies and amplitudes depend on the zeros of ... |

5 | Limiting Distributions and Zeros of Artin L-functions - Ng - 2000 |

4 |
Details of the first region of integers x with π3,2(x
- Bays, Hudson
- 1978
(Show Context)
Citation Context ... x ≥ 3. January 2006] PRIME NUMBER RACES 9How can we improve this approximation? The idea is to “add” a second wave to the first, this second wave going through two complete cycles over the interval =-=[0, 1]-=- rather than only one cycle. This corresponds to hearing the sound of the two waves at the same time, superimposed; mathematically, we literally add the two functions together. As it turns out, adding... |

3 | A common combinatorial principle underlies Riemann’s formula, the Chebyshev phenomenon, and other subtle effects in comparative prime number theory - Hudson - 1980 |

2 |
Prime territory: Exploring the infinite landscape at the base of the number system, The Sciences
- Bombieri
- 1992
(Show Context)
Citation Context ...is that we have no idea what formula similar to (3) would count prime pairs as a sum of nice “waves.” ACKNOWLEDGMENTS. The section “Doing the Wave” was inspired by Enrico Bombieri’s delicious article =-=[4]-=-. We thank Carter Bays, Kevin Ford, Richard Hudson, and Nathan Ng for making available parts of their works that have not appeared in print. We also thank Guiliana Davidoff and Michael Guy for prepari... |

2 | The prime number race and zeros of Dirichlet L-functions off the critical line - Ford, Konyagin - 2003 |

2 |
The strong law of small numbers, this Monthly 95
- Guy
- 1988
(Show Context)
Citation Context ...ery low, but it is not obvious from this limited data that they are tending towards 0. There was a wonderful article in this MONTHLY by Richard Guy some years ago, entitled “The Law of Small Numbers” =-=[11]-=-, in which Guy pointed out several fascinating phenomena that are “evident” for small integers yet disappear when one examines bigger integers. Could this be one of those phenomena? Another prime race... |

2 |
On the Shanks-Rényi race problem
- Kaczorowski
- 1996
(Show Context)
Citation Context ...cture can be generalized to the other prime races we considered.) However, the supporting evidence from our data was not entirely convincing. Indeed, by studying the explicit formula (3), Kaczorowski =-=[14]-=- and Sarnak [22], independently, showed that the Knapowski-Turán conjecture is false! In fact, the quantity 1 #{x ≤ X: there are more primes of the form 4n + 3uptoxthan of the form 4n + 1} X does not ... |

2 |
Comparative prime-number theory
- Knapowski, Turán
- 1962
(Show Context)
Citation Context ...nd the primes of the form qn + b,wherea is a square modulo q and b is not. Stark’s results thus pertained to the first case not covered by the prior work of Littlewood [18] and of Knapowski and Turán =-=[16]-=-, [17]. As he himself pointed out, it seemed particularly difficult to show that qn + a leads infinitely often, even in the case of primes of the form 5n + 4 racing against primes of the form 5n + 2. ... |

2 | Asymmetries in the Shanks-Rényi Prime Number Race - Martin |

1 |
of Dirichlet L-functions and irregularities in the distribution of primes
- Zeros
- 2000
(Show Context)
Citation Context ...ere are more primes of the form 4n + 1 up to x than primes of the form 4n + 3. In fact, there are arbitrarily large values of x for which #{primes 4n + 1 ≤ x}−#{primes 4n + 3 ≤ x} ≥ 1 √ x ln ln ln x. =-=(2)-=- 2 ln x 2 Note that this restriction is necessary, since every integer of the form qn + a is divisible by the greatest common divisor of a and q, hence cannot be prime (except possibly for a single va... |

1 |
Extensions of some results of Harold Stark on comparative prime number theory (to appear
- Davidoff, Osowski, et al.
(Show Context)
Citation Context ...e the formula analogous to the right-hand side of (3). Using Stark’s results and our own numerical calculations, we were able to prove the by now expected result in the first cases not treated by him =-=[6]-=-: Theorem. Assume that Dirichlet L-functions have no real zeros in the interval (1/2, 1). Fora= 1, 2, or4 (the squares modulo 7) and b = 3, 5, or6 (the nonsquares modulo 7) there are arbitrarily large... |

1 |
A generalization of Littlewood’s Theorem
- Davidoff
(Show Context)
Citation Context ...ctual count of primes and our approximation. Had the summer not ended, we would surely have proved that the lead changes hands infinitely often in this Mod 7 race. I was later able to go on and prove =-=[7]-=- that the lead changes hands infinitely often in the Team S vs. Team N race for any modulus q, assuming a weak version of the Generalized Riemann Hypothesis. One needs to assume only that any real zer... |

1 |
unpublished correspondence
- Sarnak
(Show Context)
Citation Context ...eralized to the other prime races we considered.) However, the supporting evidence from our data was not entirely convincing. Indeed, by studying the explicit formula (3), Kaczorowski [14] and Sarnak =-=[22]-=-, independently, showed that the Knapowski-Turán conjecture is false! In fact, the quantity 1 #{x ≤ X: there are more primes of the form 4n + 3uptoxthan of the form 4n + 1} X does not tend to any limi... |

1 |
A problem in comparative prime number theory, Acta. Arith 68
- Stark
- 1971
(Show Context)
Citation Context ...ed a generous response from Andrew Odlyzko, who pointed us to relevant data of Robert Rumely. This allowed us to begin our own work in earnest. We were most fascinated by a 1971 paper of Harold Stark =-=[25]-=-, in which he suggested a method to study prime races between primes from any two given arithmetic progressions. Back then, there was no general procedure known to prove that each team took the lead i... |

1 | On the distribution function of the remainder term of the prime number theorem - Winter - 1941 |