## Higher correlations of divisor sums related to primes, II: Variations of . . . (2007)

Citations: | 28 - 6 self |

### BibTeX

@MISC{Goldston07highercorrelations,

author = {D. A. Goldston and C. Y. Yildirim},

title = { Higher correlations of divisor sums related to primes, II: Variations of . . . },

year = {2007}

}

### OpenURL

### Abstract

We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.

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Citation Context ...og N )r ∑ a1,a2,... ,ar ai≥1, ∑ ai=k ( k a1, a2, . . . , ar ) .4 D. A. GOLDSTON AND C. Y. YILDIRIM { } k Letting denote the Stirling numbers of the second type, then it may be r easily verified (see =-=[12]-=-) that ∑ ( ) { } k k (1.14) = r! . a1, a2, . . . , ar r a1,a2,... ,ar ai≥1, ∑ ai=k We conclude that for h ∼ λlog N, (1.15) Mk(N, h, ψ) ∼ N(log N) k k∑ r=1 { k r } λ r , which are the moments of a Pois... |

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Citation Context ...s more than sufficient. In this case there have been a string of improvements. For ease of comparison with the value B = 4 used above, the value B = 3.5 obtained by Bombieri, Friedlander, and Iwaniec =-=[2]-=- gives the values Ξ1 ≤ 0.43493 . . . , Ξ2 ≤ 1.39833 . . . , Ξ3 ≤ 2.38519 . . . , Ξ4 ≤ 3.37842 . . . . All of these results above actually hold for a positive percentage of gaps. Maier [22] introduced ... |

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Citation Context ...< θk < ϑ k−1 for 2 ≤ k ≤ 3, we have, ˜M1(N, h, ψR) ∼ λN log N, ˜M2(N, h, ψR) ∼ (θ2λ + λ 2 )N log 2 N, ˜M3(N, h, ψR) ∼ (θ3 2 λ + 3θ3λ 2 + λ 3 )N log 3 N. The starting point of Bombieri and Davenport’s =-=[1]-=- work on small gaps between primes is essentially equivalent to the inequality (1.33) Letting (1.34) 2N∑ n=N+1 ((ψ(n ) ( + h) − ψ(n) − ψR(n + h) − ψR(n) )) 2 ≥ 0. M ′ k (N, h, ψ) = Mk(2N, h, ψ) − Mk(N... |

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differences between prime numbers
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Citation Context ... and Iwaniec [2] gives the values Ξ1 ≤ 0.43493 . . . , Ξ2 ≤ 1.39833 . . . , Ξ3 ≤ 2.38519 . . . , Ξ4 ≤ 3.37842 . . . . All of these results above actually hold for a positive percentage of gaps. Maier =-=[22]-=- introduced a new method to prove that ( ) pn+1 − pn liminf ≤ e n→∞ log pn −γ (1.43) = 0.56145 . . . . This method, which applies to special sets of sparse intervals, may be combined with the earlier ... |

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Citation Context ...1.3) Λ(n)Λ(n + k) = |S(α)| 2 e(−kα)dα + O(k log 2 N), where n≤N 0 S(α) = ∑ Λ(n)e(nα), e(u) = e 2πiu . n≤N For α close to the rational number a/r, we write α = a/r + β, and approximate S(α) throughout =-=[0, 1]-=- by a sum of local approximations ∑ ∑ r≤R 1≤a≤r (a,r)=1 But the last expression is equal to ∑ µ(r) φ(r) I(a ∑ + β), where I(u) = e(nu). r n≤N n≤N λR(n)e(nβ), suggesting we replace Λ(m) by λR(m) in sum... |

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Citation Context ...orems 1.6 and 1.7) of [7] on primes in short intervals, in particular, lim inf n→∞ � � pn+r − pn log pn � r − 1√ 2 r. (1.46) For longer intervals, instead of (1.42) we shall have recourse to Hooley’s =-=[12]-=- bound depending on GRH that, for all q � x, � 1�a�q (a,q)=1 max u�x |E(u;q,a)|2 ≪ x(log x) 4 . (1.47)s206 D. A. GOLDSTON AND C. Y. YILDIRIM It is easy to see that the same bound holds when the sum is... |

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Citation Context ...˜S1(N,j, a)= Λ(n + j1)=ψ(N)+O(|j1|log N) ∼ N (1.6) n=1 for |j1| = o(N/log N) by the prime number theorem (as usual ψ(x)= � n�x Λ(n)). For k = 1 and k = 2 these correlations have been evaluated before =-=[4, 13, 14]-=-, and the more general cases of n running through arithmetic progressions were also worked out [6, 11, 14]. Correlations which include in their summands factors such as Λ(n)Λ(n + j), with j �=0, canno... |

10 |
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Citation Context ...ms of the form � a(R,r) with a(R,1)=1,a(R,r) ∈ R, r�R r|n λR(n) is the best approximation to Λ(n) inanL2 sense. The proof involves a minimization which was solved in a more general setting by Selberg =-=[17]-=- for his upper bound sieve. Hooley’s recent use of λR(n)in[13, 14] leans greatly on its origin in the Selberg sieve. It should further be mentioned that, as far as we know, Heath-Brown [11] was the fi... |

10 |
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Citation Context ...ecial cases of Lemma 2.1 have been proved before. When j = 0 this was used by Selberg [25], and also Graham [13], but we have made the error term stronger with regard to k by an argument suggested in =-=[5]-=-. It is easy to make the j dependence explicit in the error term, but in this paper we will assume j is fixed (actually we only use j ≤ 2.) We will sometimes use Lemma 2.1 in the weaker form ∑ d≤R (d,... |

9 |
On a result of
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Citation Context ...α m 2 at all prime powers, with constants α1 > 0 and 0 <α2 < 2, then we have, uniformly for x � 2, � f(n) ≪α1,α2 n�x p|n x � exp log x p�x f(p) . (2.5) p This result, quoted from [9] (which refers to =-=[10]-=- for the proof of a sharper version), helps us see that for monic polynomials Pi, � P1(p) P2(p) p|n behaves on average the same as ndeg P1−deg P2 . In particular, we have � � P1(p) P2(p) n�x p|n ≪ x (... |

9 |
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Citation Context ...B N for some positive constant B > 1. Our method can also be used to examine larger than average gaps between primes. In this case much more is known than for small gaps; the latest result being that =-=[24]-=- prove (1.47) for θ > 1 4 (1.50) pn+1 − pn max ≥ (2e pn≤N log pn γ log log N log log log log N − o(1)) (log log log N) 2 . If one were to ask however for a positive proportion of gaps larger than the ... |

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(Show Context)
Citation Context ...=1 Assuming the Hardy–Littlewood conjecture in the strong form ψ j (x)=S(j)x + O(N 1/2+ɛ ) (1.20) uniformly for 1 � r � k, 1� x � N and distinct ji satisfying 1 � ji � h, Montgomery and Soundararajan =-=[16]-=- proved that � μk(N, h, ψ) ∼ (1 · 3 · ...· (k − 1))N h log N h μk(N, h, ψ) ≪ N(h log N) k/2 � �−1/(8k) h log N � k/2 + h k N 1/2+ɛ if k is even, (1.21) if k is odd, (1.22) uniformly for (log N) 1+δ � ... |

7 | Yildirim, Small gaps between primes exist - Goldston, Motohashi, et al. |

7 |
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Citation Context ...k ) can be improved. This has been done in the case k = 2 (unpublished) where the error term O(R 2 ) may be replaced by O(R 2−δ ) for a small constant δ. In the special case of S2(N, (0), (2)) Graham =-=[13]-=- has removed the error term O(R 2 ) entirely. In proving Theorem 1.1 we will assume j1 = 0. This may be done without loss of generality since we may shift the sum over n in Sk to m = n + j1 and then r... |

6 |
L.K.: Introduction to Number Theory, translated from the Chinese by Peter Shiu
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Citation Context ...e1)d(e2)...d(em) d(e1)d(e2)...d(em) [e1, e2, . . . , em] ∑ n≤N ei|n,1≤i≤m + O 1 ( ( ∑ e≤R d(e) ) m ) . The last error term is O(Rm log m R), and DR(n) ≤ ∑ e|n d(n) = d2 (n). Hence, using the estimate =-=[19]-=- ∑ d(m) k ≪k N log 2k−1 (4.14) N, we have Thus m≤N ∑ DR(n) m ≤ ∑ d(n) 2m ≪m N log 4m−1 N. n≤N ∑ d1,d2,... ,dm≤R n≤N d(Dm) ≤ Dm ∑ e1,e2,... ,em≤R d(e1)d(e2)...d(em) [e1, e2, . . . , em] ≪m log 4m−1 R N... |

5 |
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(Show Context)
Citation Context ...by Selberg [17] for his upper bound sieve. Hooley’s recent use of λR(n)in[13, 14] leans greatly on its origin in the Selberg sieve. It should further be mentioned that, as far as we know, Heath-Brown =-=[11]-=- was the first to use λR(n) in additive prime number theory. The correlations that we are interested in evaluating are N� Sk(N,j, a)= λR(n + j1) a1 λR(n + j2) a2 ...λR(n + jr) ar (1.4) and ˜Sk(N,j, a)... |

5 |
On the Goldbach’s problem and the sieve methods, Sci. Sinica 21
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(Show Context)
Citation Context ... (1.42) is equal to r − 1 + 1 B 2 + O(1 r ), and thus for large r this bound approaches r − 5 8 with B = 4. The best result known for B which holds uniformly for all k is B = 3.9171 . . . due to Chen =-=[4]-=-. However, in the application to obtain (1.42) one only needs (1.39) to hold uniformly for a restricted range of k; the condition 0 < |k| ≤ log 2 N is more than sufficient. In this case there have bee... |

5 |
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(Show Context)
Citation Context ...ile (1.35) has never been improved, the refinements based on the Erdös method together with the choice of certain weights in a more general form of (1.35) has led to further improvements. Huxley [20] =-=[21]-=- proved that, letting θr be the smallest positive solution of θr + sin θr = π Br , sin θr (1.41) < (π + θr)cosθr , then (1.42) Ξr ≤ With the value B = 4 this gives 2r − 1 4Br { Br + (Br − 1) θr } . si... |

4 |
Über die punktweise Konvergenz von Ramanujan-Entwicklungen zahlentheoretischer Funktionen, Acta Arithmetica XLIV
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(Show Context)
Citation Context ...,jr−1,ji) and ai =(a1,a2,...,ai−1,ai+1,...,ar−1,1), and Lk(R):= � r�R (r,k)=1 μ2 (r) . (1.39) φ(r) (It is easily seen that L1(R) ≪ log 2R for R � 1. A precise estimation of this sum due to Hildebrand =-=[9]-=- is given in (2.15) below.) Formula (1.38) reduces the calculation of the mixed moments to the calculation of mixed correlations. Our results depend on the extent of uniformity in the distribution of ... |

4 |
The Riemann Zeta-function (translated from the Russian by N. Koblitz), Walter de Gruyter
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(Show Context)
Citation Context ... δ + it)|3 √ dt δ2 + t2 ≪ x δ 1 ( ( 1 2 − δ)2 + log3 (3.25) T), if we take 1 U 2 − log 2 3 T 1 < δ < 2 , U being an appropriate constant. To see this we employ the estimate (see Karatsuba and Voronin =-=[16]-=-, p.116) (3.26) ζ(σ + it) = O(log 2 3 |t|), (σ ≥ 1 − U which implies (3.27) x δ ∫ T |ζ( 1 2 2 + δ + it)|3 √ dt ≪ x δ2 + t2 δ ∫ T 2 log 2 3 |t| log 2 t t , |t| ≥ 2), dt ≪ x δ log 3 T.HIGHER CORRELATIO... |

4 |
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(Show Context)
Citation Context ...OLDSTON AND C. Y. YILDIRIM convenience we define (1.5) ˜S1(N, j, a) = N∑ Λ(n + j1) ∼ N n=1 with |j1| ≤ N by the prime number theorem. For k = 1 and k = 2 these correlations have been evaluated before =-=[8]-=- (and for λQ(n) they have been evaluated in [9]); the results show that ΛR and λR mimic the behavior of Λ, and this is also the case in arithmetic progressions, see [17], [18], [11]. When k ≥ 3 the pr... |

4 | Méthodes de crible appliquées aux sommes de Kloosterman et aux petits écarts entre nombres premiers, Thése de Doctorat de l’Université - Sivak - 2005 |

3 |
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(Show Context)
Citation Context ...(α)| 2 e(−kα) dα + O(k log 2 N), (1.3) where n�N r�R 1�a�r (a,r)=1 0 S(α)= � Λ(n)e(nα), e(u)=e 2πiu . n�N For α close to the rational number a/r, we write α = a/r + β, and approximate S(α) throughout =-=[0,1]-=- by a sum of local approximations � � μ(r) φ(r) I � a � + β , where I(u)= r � e(nu). However, the above double sum is equal to � n�N λR(n)e(nβ), which suggests that we replace Λ(m) byλR(m) in sums suc... |

3 | On the second moment for primes in an arithmetic progression
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(Show Context)
Citation Context ...as usual ψ(x)= � n�x Λ(n)). For k = 1 and k = 2 these correlations have been evaluated before [4, 13, 14], and the more general cases of n running through arithmetic progressions were also worked out =-=[6, 11, 14]-=-. Correlations which include in their summands factors such as Λ(n)Λ(n + j), with j �=0, cannot be evaluated unconditionally; they are the subject of the Hardy–Littlewood prime rtuple conjecture [8]. ... |

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2 |
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(Show Context)
Citation Context ...r r odd (1.26) (A =2− γ − log 2π, and γ denotes Euler’s constant). Gallagher’s result (1.17) can be deduced from these. Note that it is easy to see that R1(h) = 0, and for r = 2 we know from Goldston =-=[3]-=- that (1.25) holds with the much smaller error term O(h1/2+ɛ ). Only the first moment is known unconditionally as a simple consequence of the prime number theorem. The work of Goldston and Montgomery ... |

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2 | Longer than average intervals containing no primes
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(Show Context)
Citation Context ...an average. To formalize this, we let Θr be the supremum over all λ for which (1.51) ∑ N<pn≤2N pn+r−pn≥λlog N (pn+r − pn) ≫λ N for all sufficiently large N. Then using the Erdös method one finds that =-=[3]-=- Θ1 ≥ 1 + 1 (1.52) 2B where B is the number in (1.39). To apply (1.44) to this problem, we assume that pj+r − pj < h = λlog N for all N < pj ≤ 2N in which case the interval (n, n + h] always contains ... |

2 | 771? second moment for prime numbers - Goldston - 1984 |

2 |
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(Show Context)
Citation Context ... proved. It is known that the asymptotic formula for the second moment follows from the assumption of the Riemann Hypothesis and the pair correlation conjecture for zeros of the Riemann zeta-function =-=[10]-=-. Turning to our approximation ΛR(n), we define ψR(x) = ∑ (1.16) ΛR(n) n≤x and first wish to examine the moments M(N, h, ψR) defined as in (1.6). The same computation used for ψ to obtain (1.10) clear... |

2 |
Beyond pair correlation to appear
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(Show Context)
Citation Context ...lly be considering the case λ ≪ 1. When h is larger we need to subtract the expected value h in the moments above, which leads to more delicate questions which we will not consider in this paper (see =-=[23]-=-). Gallagher [6] proved that the moments in (1.6) may be computed from the Hardy-Littlewood prime rtuple conjecture [14]. This conjecture states that for j = (j1, j2, . . . , jr) with the ji’s distinc... |

1 | Sieves in number theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 43 - Greaves - 2001 |

1 |
A lower bound method for binary additive problems involving primes, Unpublished manuscript
- Goldston
(Show Context)
Citation Context ...elate λR(n) to ΛR(n) by interchanging the order of the summations in (1.1), thereupon the new inner sum can be evaluated (eq. (2.15) below) and the contribution of its main term gives ΛR(n). Goldston =-=[5]-=- found λR(n) while remedying the failure of the circle method in an application to the related problems of twin primes and short gaps between primes for which a starting point is the observation that ... |