### Abstract

Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define S(x) = ∑_{n≤x} r(n) and (x) = S(x) - αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of ∫_{0}^{∞} (e^{y})^{2} e^{-2yσ} dy is 1/3 as expected.

Original language | English (US) |
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Pages (from-to) | 169-183 |

Number of pages | 15 |

Journal | Journal of Number Theory |

Volume | 100 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*100*(1), 169-183. https://doi.org/10.1016/S0022-314X(02)00075-6

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*Journal of Number Theory*, vol. 100, no. 1, pp. 169-183. https://doi.org/10.1016/S0022-314X(02)00075-6

**A mean value theorem for cubic fields.** / Vaughan, Robert Charles.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A mean value theorem for cubic fields

AU - Vaughan, Robert Charles

PY - 2003/5/1

Y1 - 2003/5/1

N2 - Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define S(x) = ∑n≤x r(n) and (x) = S(x) - αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of ∫0∞ (ey)2 e-2yσ dy is 1/3 as expected.

AB - Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define S(x) = ∑n≤x r(n) and (x) = S(x) - αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of ∫0∞ (ey)2 e-2yσ dy is 1/3 as expected.

UR - http://www.scopus.com/inward/record.url?scp=0038368022&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038368022&partnerID=8YFLogxK

U2 - 10.1016/S0022-314X(02)00075-6

DO - 10.1016/S0022-314X(02)00075-6

M3 - Article

AN - SCOPUS:0038368022

VL - 100

SP - 169

EP - 183

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -