## Theorem

### BibTeX

@MISC{Heath-brown_theorem,

author = {D. R. Heath-brown},

title = {Theorem},

year = {}

}

### OpenURL

### Abstract

in celebration of his Sixtieth Birthday Let d be a square-free integer, which may be positive or negative, and let h(−d) be the class number of Q ( √ −d). In this paper we investigate the frequency of values of d for which 3|h(−d). It follows from conjectures of Cohen and Lenstra [3], that asymptotically a constant proportion of values of d have this property. The conjectured proportion is different for positive and negative d, being 1 − (1 − 3 −j) j=1 in the case of imaginary quadratics, for example. It follows from the work of Davenport and Heilbronn [5] that a positive proportion of d have 3 ∤ h(−d), both in the case of d positive and d negative. However it remains an open problem whether or not the same is true for values with 3|h(−d). Write N−(X) for the number of positive square-free d ≤ X for which 3|h(−d), and similarly let N+(X) be the number of positive square-free d ≤ X for which 3|h(d). It was shown by Ankeny and Chowla [1] that N−(X) tends to infinity with X, and in fact their method yields N−(X) ≫ X 1/2. The best known result in this direction is that due to Soundararajan [7], who shows that N−(X) ≫ε X 7/8−ε, for any positive ε. In the case of real quadratic fields it was shown by Byeon and Koh [2] how Soundararajan’s analysis can be adapted to prove N+(X) ≫ε X 7/8−ε. The purpose of this note is to present a small improvement on these results, as follows.

### Citations

33 |
Heuristics on class groups of number fields, Number theory
- Lenstra
- 1983
(Show Context)
Citation Context ...y be positive or negative, and let h(−d) be the class number of Q( √ −d). In this paper we investigate the frequency of values of d for which 3|h(−d). It follows from conjectures of Cohen and Lenstra =-=[3]-=-, that asymptotically a constant proportion of values of d have this property. The conjectured proportion is different for positive and negative d, being ∞∏ 1 − (1 − 3 −j ) j=1 in the case of imaginar... |

21 |
The Theory of the Riemann Zeta-Function (2nd edition, revised by D.R. HeathBrown), Oxford Science
- Titchmarsh
- 1986
(Show Context)
Citation Context ... ∈ (0, 1 4 ). The proof of (1), and hence of (2), depends on the definition of βν, so we must check that our modification does not materially alter the estimates. It is only Lemma 10.12 of Titchmarsh =-=[4]-=- which needs any change. It is shown that ∑ κ≤X/d ακκ θ−1 log X ≪ (X dκ d )θ(log X d ∏ )1/2 (1 + p −1 ) 1/2 p|ρ (3) 2uniformly for 0 < θ ≤ 1 2 , where κ is restricted to integers coprime to ρ. We sha... |

13 |
There are infinitely many Carmichael
- Alford, Granville, et al.
- 1994
(Show Context)
Citation Context ...ny prime triplets of this form. Indeed it is not known whether or not there are infinitely many Carmichael numbers with 3 prime factors. None the less it was proved by Alford, Granville and Pomerance =-=[1]-=- that there are infinitely many Carmichael numbers. Let C3(x) denote the number of Carmichael numbers n ≤ x having ω(n) = 3. It has been conjectured by Granville and Pomerance [3] that C3(x) ∼ c x1/3 ... |

12 | Two contradictory conjectures concerning Carmichael numbers
- Granville, Pomerance
(Show Context)
Citation Context ...nville and Pomerance [1] that there are infinitely many Carmichael numbers. Let C3(x) denote the number of Carmichael numbers n ≤ x having ω(n) = 3. It has been conjectured by Granville and Pomerance =-=[3]-=- that C3(x) ∼ c x1/3 , (log x) 3 with an explicit positive constant c. Various upper bounds approximating this have been given, the best available in the literature being the estimate C3(x) ≪ε x 5/14+... |

10 | S.V.Nagaraj, Density of Carmichael numbers with three prime factors
- Balasubramanian
- 1997
(Show Context)
Citation Context ...onstant c. Various upper bounds approximating this have been given, the best available in the literature being the estimate C3(x) ≪ε x 5/14+ε , for any fixed ε > 0, due to Balasubramanian and Nagaraj =-=[2]-=-. The goal of the present paper is to improve this as follows. Theorem For any fixed ε > 0 we have C3(x) ≪ε x 7/20+ε . Note that 1/3 < 7/20 < 5/14. Indeed 7 20 5 14 − 1 3 − 1 3 = 7 10 . Thus our resul... |

10 |
On the uniformity of the distribution of the zeros of the Riemann zeta function
- Fujii
- 1978
(Show Context)
Citation Context ...r ordinates γi of the zeros in the usual sense, not restricting to those zeros which are on the critical line, then one has ∑ γi≤T (γi+1 − γi) µ ≪µ T (log T ) 1−µ for any µ > 0, as was shown by Fujii =-=[1]-=-. In giving the proof of our result we shall refer to the version of Selberg’s argument presented by Titchmarsh [4: §§10.9-10.22]. The proof uses a “mollifier” φ(s) = ∑ βνν −s , ν≤X 1in which the num... |

7 |
On the divisibility of the class number of quadratic fields
- Ankeny, Chowla
- 1955
(Show Context)
Citation Context ...d). Write N−(X) for the number of positive square-free d ≤ X for which 3|h(−d), and similarly let N+(X) be the number of positive square-free d ≤ X for which 3|h(d). It was shown by Ankeny and Chowla =-=[1]-=- that N−(X) tends to infinity with X, and in fact their method yields N−(X) ≫ X 1/2 . The best known result in this direction is that due to Soundararajan [7], who shows that N−(X) ≫ε X 7/8−ε , for an... |

7 | On the zeros of Riemann's zeta-function - Selberg - 1942 |

2 |
Real quadratic fields with class number divisible by 3
- Byeon, Koh
- 1983
(Show Context)
Citation Context .../2 . The best known result in this direction is that due to Soundararajan [7], who shows that N−(X) ≫ε X 7/8−ε , for any positive ε. In the case of real quadratic fields it was shown by Byeon and Koh =-=[2]-=- how Soundararajan’s analysis can be adapted to prove N+(X) ≫ε X 7/8−ε . The purpose of this note is to present a small improvement on these results, as follows. Theorem For large X we have N−(X) ≫ε X... |

1 |
J.E.: Contributions to the theory of the Rieman zeta-function and the theory of the distribution of primes
- Hardy, Littlewood
- 1918
(Show Context)
Citation Context ... ≤ a ′ (K), say. We are now ready to estimate ∑ ˆγi (ˆγi+1 − ˆγi) µ for T ≤ ˆγi ≤ 2T. We shall choose K to be a fixed integer such that 2 < 2 − µ. 2K + 1 According to a result of Hardy and Littlewood =-=[2]-=- we have ˆγi+1 − ˆγi ≪ T θ for any θ > 1 4 , so that it suffices to prove that ∑ (ˆγi+1 − ˆγi) µ ≪ T (log T ) 1−µ (11) ˆγi for T ≤ ˆγi < ˆγi+1 ≤ 2T. Moreover, summands for which ˆγi+1 − ˆγi ≤ 8 a ′ (K... |