## On the second moment for primes in an arithmetic progression, Acta Arithmetica C.1 (2001)

Citations: | 3 - 2 self |

### BibTeX

@MISC{Goldston01onthe,

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

title = {On the second moment for primes in an arithmetic progression, Acta Arithmetica C.1},

year = {2001}

}

### OpenURL

### Abstract

Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together with some new results for binary correlations of this divisor sum approximation in arithmetic progressions. 1. Introduction and Statement of

### Citations

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Citation Context ...e, so the assumption (1.10) is a way of expressing that the zeros of different Dirichlet L-functions are uncorrelated. Theorem C is a generalization of one half of a result of Goldston and Montgomery =-=[4]-=- for the case q = 1, where an equivalence between the pair correlation conjecture for ζ(s) and the second moment for primes was established. Since the argument in [4] works reversibly, a suitable conv... |

16 |
On the Barban-Davenport-Halberstam theorem XIII
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Citation Context ...sum is S(k) and the error term is ≪ the O-term in (3.9). Lemma 7. (Goldston and Friedlander [1])We have ∑ 0<j≤ h q (h − jq)S(jq) = h2 h h − log 2φ(q) 2 q + O(h(log log 3q)3 ). (3.14) Lemma 8. (Hooley =-=[10]-=-) Assuming GRH, we have ∑ a(mod q) (a,q)=1 max u≤x |E(u; q, a)|2 ≪ x log 4 x; for q ≤ x. (3.15) Lemma 9. We have ∑ r≤R ∑ r≤R µ 2 (r)σ(r) φ(r) ≪ R (3.16) µ 2 (r)σ 1 2 (r) φ(r) ≪ √ R (3.17) ∑ 0<r≤R rd(r... |

10 |
A lower bound for the second moment of primes in short intervals, Expo
- Goldston
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Citation Context ... the Generalized Riemann Hypothesis (GRH), which implies, in particular, E(x; q, a) := ψ(x; q, a) − x 1 ≪ x 2 log φ(q) 2 x , (q ≤ x). (1.4) The idea of our method originates from the work of Goldston =-=[3]-=- for the case of all primes, corresponding in the present formulation to q = 1. An improved and generalized version of this result appeared in [5] as 1991 Mathematics Subject Classification. Primary:1... |

8 | On the pair correlation of zeros of the Riemann zeta-function
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Citation Context ... dµ(d). (3.1) This function is known ([3], [8]) to exhibit behavior similar to Λ(n) when considered on average in arithmetic progressions, and it has been employed in related problems ([1], [2], [5], =-=[6]-=-). An upper bound for λR(n) is |λR(n)| ≤ ∑ d ∑ d|n r≤R d|r 1 φ(r) r ∑ ≤ max( ) r≤R φ(r) d|n d ∑ r≤R d|r 1 r ≪ d(n) logRlog log R. (3.2) To evaluate the sums which arise when (3.1) is used in (2.6) and... |

7 | An asymptotic estimate related to Selberg's sieve - Graham - 1978 |

6 | Variance of distribution of primes in residue classes
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Citation Context ... φ(r) ∑ d|(r,n) dµ(d). (3.1) This function is known ([3], [8]) to exhibit behavior similar to Λ(n) when considered on average in arithmetic progressions, and it has been employed in related problems (=-=[1]-=-, [2], [5], [6]). An upper bound for λR(n) is |λR(n)| ≤ ∑ d ∑ d|n r≤R d|r 1 φ(r) r ∑ ≤ max( ) r≤R φ(r) d|n d ∑ r≤R d|r 1 r ≪ d(n) logRlog log R. (3.2) To evaluate the sums which arise when (3.1) is us... |

5 |
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Citation Context ...b) (2.7)6 GOLDSTON AND YILDIRIM 3. The choice of λR(n) and some number-theoretic sums As the auxiliary function we use λR(n) := ∑ r≤R µ 2 (r) φ(r) ∑ d|(r,n) dµ(d). (3.1) This function is known ([3], =-=[8]-=-) to exhibit behavior similar to Λ(n) when considered on average in arithmetic progressions, and it has been employed in related problems ([1], [2], [5], [6]). An upper bound for λR(n) is |λR(n)| ≤ ∑ ... |

4 |
ÄUber die punktweise Konvergenz von Ramanujan-Entwicklungen zahlentheoretischer Funktionen, Acta Arithmetica XLIV
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Citation Context ...r 1 r ≪ d(n) logRlog log R. (3.2) To evaluate the sums which arise when (3.1) is used in (2.6) and (2.7) we shall need some lemmas. In the following p will denote a prime number. Lemma 3. (Hildebrand =-=[9]-=-) We have for each positive integer k, uniformly in R ≥ 1, Lk(R) := ∑ n≤R (n,k)=1 µ 2 (n) φ(n) φ(k) = (log R + c + v(k)) + O(w(k) √ ), (3.3) k R where c :=γ + ∑ p log p p(p − 1) ; ∑ v(k) := p|k log p ... |

3 |
The pair correlation of zeros of Dirichlet L-functions and primes in arithmetic progressions
- Yıldırım
- 1991
(Show Context)
Citation Context ....7) 2 φ(q) We shall see below that the GRH implies a result of the type in Theorem B for much wider ranges of q and h. An asymptotic estimate for I(x, h, q, a) in certain ranges was shown by Yıldırım =-=[12]-=- to be implied by GRH and a pair correlation conjecture for the zeros of Dirichlet’s L-functions. Theorem C. Assume GRH. Let α1, α2, η be fixed and satisfying 0 < η < α1 ≤ α2 ≤ 1, and let δ = x −α whe... |

2 | Yıldırım, Primes in short segments of arithmetic progressions
- Goldston, Y
- 1998
(Show Context)
Citation Context ...ea of our method originates from the work of Goldston [3] for the case of all primes, corresponding in the present formulation to q = 1. An improved and generalized version of this result appeared in =-=[5]-=- as 1991 Mathematics Subject Classification. Primary:11M26. 1 Research at MSRI is supported in part by NSF grant DMS-9701755, also supported by an NSF grant 2 Research at MSRI is supported in part by ... |

1 |
Note on a variance in the distribution of primes, Number Theory
- Friedlander, Goldston
- 1997
(Show Context)
Citation Context ... ∑ d|(r,n) dµ(d). (3.1) This function is known ([3], [8]) to exhibit behavior similar to Λ(n) when considered on average in arithmetic progressions, and it has been employed in related problems ([1], =-=[2]-=-, [5], [6]). An upper bound for λR(n) is |λR(n)| ≤ ∑ d ∑ d|n r≤R d|r 1 φ(r) r ∑ ≤ max( ) r≤R φ(r) d|n d ∑ r≤R d|r 1 r ≪ d(n) logRlog log R. (3.2) To evaluate the sums which arise when (3.1) is used in... |

1 |
On the irregularity of distribution of primes in an arithmetic progression over short intervals
- Özlük
- 1987
(Show Context)
Citation Context ... xhlog Moreover, for almost all q with h 3/4 log 5 x ≤ q ≤ h we have we have ((q ) ) 3x − O(xh(log log x) h 3 ). (1.5) I(x, h, q) ∼ xhlog( xq ). (1.6) h For an individual arithmetic progression Özlük =-=[11]-=- proved unconditionally Theorem B. For 1 ≤ q ≤ (log x) 1−δ , and h ≤ (log x) c (δ and c are any fixed positive numbers) satisfying q ≤ h, we have for any ǫ and x ≥ X(ǫ, c). I(x, h, q, a) > ( 1 xh − ǫ)... |