A mean value theorem for cubic fields

R. C. Vaughan

doi:10.1016/S0022-314X(02)00075-6

A mean value theorem for cubic fields

R. C. Vaughan

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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)
Pages (from-to)	169-183
Number of pages	15
Journal	Journal of Number Theory
Volume	100
Issue number	1
DOIs	https://doi.org/10.1016/S0022-314X(02)00075-6
State	Published - May 1 2003

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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

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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∞ (ey)2 e-2yσ dy is 1/3 as expected.",

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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.

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JO - Journal of Number Theory

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A mean value theorem for cubic fields

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