## Primes in short intervals

Venue: | Commun. Math. Phys |

Citations: | 8 - 3 self |

### BibTeX

@ARTICLE{Montgomery_primesin,

author = {Hugh L. Montgomery and K. Soundararajan},

title = {Primes in short intervals},

journal = {Commun. Math. Phys},

year = {},

pages = {589--617}

}

### OpenURL

### Abstract

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

### Citations

133 |
Some problems on partitio numerorum III : On the expression of a number as a sum of primes
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- 1923
(Show Context)
Citation Context ...log N/H. Certainly Conjecture 2 does not hold when H ≍ N, but perhaps it holds whenever H = o(N). It would be interesting to investigate more thoroughly what happens in this range. Hardy & Littlewood =-=[10]-=- provided heuristics that point toward the quantitative prime k-tuple conjecture (1). In §4 we argue in the same spirit to obtain indications in favor of Conjecture 1. To obtain further support for ou... |

81 | On the distribution of spacings between zeros of the zeta function
- Odlyzko
- 1987
(Show Context)
Citation Context ...of this distribution is 0 if k is odd, and is ∼ µk(N/2) k/2 if k is even. As for numerical studies, Brent [2] has compiled evidence not only for (1) but also for the stronger hypothesis (20). Odlyzko =-=[18]-=- and Forrester & Odlyzko [5] have found that the local distribution of the zeros of the zeta function fits well with predictions based on random matrix theory. The authors [15] have reported on numeri... |

70 |
The pair correlation of zeros of the zeta function. In Analytic number theory (Proc
- Montgomery
- 1972
(Show Context)
Citation Context ...eros of the Riemann zeta function. We recall that Goldston & Montgomery [8] showed that if RH is true, then the stronger form (F(α) ∼ 1) of the Pair Correlation Conjecture as formulated by Montgomery =-=[13]-=- is equivalent to the case K = 2 of the Conjecture above. In the same spirit, Chan [3] has shown (assuming RH) that Conjecture 1 is equivalent to the assertion that (22) ∫ X 1 ( ∑ 0<γ≤T ) k ( T ) k/2 ... |

44 |
On the order of magnitude of the difference between consecutive prime numbers
- Cramér
- 1936
(Show Context)
Citation Context ...ime 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 . 0. Introduction 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 pro... |

32 |
Primes in short intervals
- Maier
(Show Context)
Citation Context ...(q)/q n=1 m=1 (m+n,q)=1 be the kth centered moment of the number of reduced residues (mod q) in an interval. Lemma 2 of Montgomery & Vaughan asserts that ( φ(q) ) k (11) mk(q; h) = q Vk(q; h) q where =-=(12)-=- Vk(q; h) = ∑ d1,... ,dk 1≤di≤h ∑ q1,... ,qk 1<qi|q ( k ∏ i=1 aidi qi ) µ(qi) ∑ ( ∑k e φ(qi) a1,... ,ak i=1 1≤ai≤qi ∑(ai,qi)=1 ai/qi∈Z When k = 1, the conditions in the innermost sum cannot be fulfill... |

30 |
Random matrix theory and the Riemann zeros II: n-point correlations. Nonlinearity 9
- Bogomolny, Keating
- 1996
(Show Context)
Citation Context ...riables, and Conjecture 1 asserts that this same sum has the same normal distribution that it would have if the terms were independent random variables. In somewhat the same vein, Bogomolny & Keating =-=[1]-=- used Hardy–Littlewood conjectures concerning primes to arrive at the n level correlation function of zeros of the zeta function. Freeman Dyson observed that the Pair Correlation Conjecture is analogo... |

27 |
Pair correlation of zeros and primes in short intervals
- Goldston, Montgomery
- 1987
(Show Context)
Citation Context ...ndications in favor of Conjecture 1. To obtain further support for our conjectures, we interpret the situation in terms of the zeros of the Riemann zeta function. We recall that Goldston & Montgomery =-=[8]-=- showed that if RH is true, then the stronger form (F(α) ∼ 1) of the Pair Correlation Conjecture as formulated by Montgomery [13] is equivalent to the case K = 2 of the Conjecture above. In the same s... |

21 |
High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107
- Rains
- 1997
(Show Context)
Citation Context ...t (22) has a similar analogue in random matrix theory. Let U(N) denote the classical compact group of unitary N × N matrices. For A ∈ U(N), let e(θ1), . . . , e(θN) denote the eigenvalues of A. Rains =-=[19]-=- has observed that if M is an integer, |M| ≥ N, then the point (Mθ1, . . . , MθN) is exactly uniformly distributed in T N as A varies with respect to the Haar measure dA on U(N). It follows in particu... |

14 |
On the distribution of primes in short intervals
- Gallagher
- 1976
(Show Context)
Citation Context ...th probability 1 − 1/ log n. If pn denotes the n th prime number this model predicts that 1 lim N→∞ N card{n : 1 ≤ n ≤ N, pn+1 − pn > c log pn} = e −c for all fixed positive real numbers c. Gallagher =-=[6]-=- showed that the above follows from Hardy & Littlewood’s [10, p. 61] quantitative version of the prime k-tuple conjecture: If D = {d1, d2, . . . , dk} is a set of k distinct integers, then (1) ∑ n≤x k... |

13 |
Irregularities in the distribution of primes and twin primes
- Brent
- 1975
(Show Context)
Citation Context ...ith mean 0 and variance N/2. By an easy calculation it can also be shown that the k th moment of this distribution is 0 if k is odd, and is ∼ µk(N/2) k/2 if k is even. As for numerical studies, Brent =-=[2]-=- has compiled evidence not only for (1) but also for the stronger hypothesis (20). Odlyzko [18] and Forrester & Odlyzko [5] have found that the local distribution of the zeros of the zeta function fit... |

11 |
Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis
- Ten
- 1994
(Show Context)
Citation Context ...bi/ri) ∣ i=1 ≤ 1 r k∏ ( i=1 ri ri ∑ bi=1 |Gi(bi/ri)| 2 ) 1/2 . Montgomery & Vaughan [17] have derived several variants of the above; an exposition of such variants is found in Chapter 8 of Montgomery =-=[14]-=-. For our present purposes a different type of variant is useful. Lemma 2. Let q1, . . . , qk be squarefree integers, each one strictly greater than 1, and put d = [q1, . . . , qk]. Let G be a complex... |

6 | An uncertainty principle for arithmetic sequences
- Granville, Soundararajan
(Show Context)
Citation Context ... times in the above sum. Thus the above asserts that the mean value of S(D) tends to 1 as h → ∞. Correspondingly, we need to estimate the quantities (8) Rk(h) = ∑ S0(D) . From (2) and (5) we see that =-=(9)-=- S0(D) = ∑ q1,... ,qk 1<qi<∞ ( k ∏ i=1 d1,... ,dk 1≤di≤h didistinct ) µ(qi) ∑ ( ∑k e φ(qi) a1,... ,ak i=1 1≤ai≤qi ∑(ai,qi)=1 ai/qi∈Z The task of estimating averages of this expression is quite challen... |

6 |
On the distribution of reduced residues
- Montgomery, Vaughan
- 1986
(Show Context)
Citation Context ...∑k e φ(qi) a1,... ,ak i=1 1≤ai≤qi ∑(ai,qi)=1 ai/qi∈Z The task of estimating averages of this expression is quite challenging, but our burden is substantially lightened by work of Montgomery & Vaughan =-=[16]-=- concerning a strikingly similar quantity. Let q∑ ( h∑ ) k (10) mk(q; h) = 1 − hφ(q)/q n=1 m=1 (m+n,q)=1 be the kth centered moment of the number of reduced residues (mod q) in an interval. Lemma 2 of... |

4 |
Beyond pair correlation, Paul Erdős and his mathematics
- Montgomery, Soundararajan
- 2002
(Show Context)
Citation Context ...ypothesis (20). Odlyzko [18] and Forrester & Odlyzko [5] have found that the local distribution of the zeros of the zeta function fits well with predictions based on random matrix theory. The authors =-=[15]-=- have reported on numerical evidence in support of the conjectures. Finally, Chan [3; pp. 36, 49, 63] has assembled evidence in favor of (22). Cramér’s model suggests that (23) π(x + (log x) a ) − π(x... |

4 |
On the normal density of primes in short intervals, and the difference between consecutive primes, Archiv for Mathematik og Naturvidenskab 47
- Selberg
- 1943
(Show Context)
Citation Context ...e, provided that the error term in (4) is sufficiently small and H is not too large. Theorem 3. Let Ek(x; D) be defined by the relation i=1 and suppose that ∑ n≤x k∏ Λ(n + di) = S(D)x + Ek(x; D), i=1 =-=(20)-=- Ek(x; D) ≪ N 1/2+ε uniformly for 1 ≤ k ≤ K, 0 ≤ x ≤ N, and distinct di satisfying 1 ≤ di ≤ H. Then (21) MK(N; H) = µKH K/2 ∫ N 1 (log x/H + B) K/2 dx ( + O N(log N) K/2 H K/2 ( H ) −1/(8K) + H log N ... |

3 |
Pair Correlation and Distribution of Prime Numbers
- Chan
- 2002
(Show Context)
Citation Context ... if RH is true, then the stronger form (F(α) ∼ 1) of the Pair Correlation Conjecture as formulated by Montgomery [13] is equivalent to the case K = 2 of the Conjecture above. In the same spirit, Chan =-=[3]-=- has shown (assuming RH) that Conjecture 1 is equivalent to the assertion that (22) ∫ X 1 ( ∑ 0<γ≤T ) k ( T ) k/2 cos(γ log x) dx = (µk + o(1))X log T . 4π Viewed in this way, we see that the Pair Cor... |

3 |
M.: Gaussian unitary ensemble eigenvalues and Riemann ζ function zeros: A nonlinear equation for a new statistic
- Forrester, Odlyzko
- 1996
(Show Context)
Citation Context ...k is odd, and is ∼ µk(N/2) k/2 if k is even. As for numerical studies, Brent [2] has compiled evidence not only for (1) but also for the stronger hypothesis (20). Odlyzko [18] and Forrester & Odlyzko =-=[5]-=- have found that the local distribution of the zeros of the zeta function fits well with predictions based on random matrix theory. The authors [15] have reported on numerical evidence in support of t... |

2 |
Linnik’s theorem on Goldbach numbers in short intervals
- Goldston
- 1990
(Show Context)
Citation Context ... if k is even, and µk = 0 if k is odd. Here the main term is the k th moment of a normal random variable with expectation 0 and variance V2(q; h). We remark that the work of Granville & Soundararajan =-=[7]-=- (see §6a) places restrictions on the uniformity (in k) with which (15) can possibly hold. With Theorem 1 in hand, we are able to estimate the Rk(h). Theorem 2. Let h be an integer, h > 1, and suppose... |

2 |
On the mean square distribution of primitive roots of unity
- Hausman, Shapiro
- 1973
(Show Context)
Citation Context ...9, 63] has assembled evidence in favor of (22). Cramér’s model suggests that (23) π(x + (log x) a ) − π(x) ∼ (log x) a−1 as x → ∞ with a fixed, a > 2. This, however, is known to be false, since Maier =-=[11]-=- showed that π(x + (log x) lim x→∞ a ) − π(x) (log x) a−1 ≷ 1 for any fixed a > 0 (for general results of this nature see Granville & Soundararajan [7]). Presumably (23) is valid for most x, and the e... |