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Franz Lemmermeyer
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Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had written in 1871/72. The conjecture was named after Dedekind by van der Waall in 1974.

Despite Artin's claim, the integrality of the quotient $\zeta_K(s)/\zeta(s)$ is not addressed at all by Dedekind (Hecke proved that $\zeta_K(s)$ extends to a meromorphic function on the complex plane in 1917). Dedekind proved that $\zeta_K(s)/\zeta(s)$ for pure cubic extensions may be written as a linear combination of Epstein zeta functions. Epstein proved in 1903 that these are meromorphic with simple poles in $s = 1$, and Artin realized in 1923 that this implies that the quotient is entire - I guess this explains his remark.

In any case Dedekind had nothing to do with this conjecture.

Edit. Landau [Über die Wurzeln der Zetafunktion eines algebraischen Zahlkörpers, Math. Ann. 79 (1919), 388-401] writes on p. 390: "I have to be careful with its formulation since it is not known whether $\zeta_k(s)/\zeta(s)$ is always an entire function".

Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had written in 1871/72. The conjecture was named after Dedekind by van der Waall in 1974.

Despite Artin's claim, the integrality of the quotient $\zeta_K(s)/\zeta(s)$ is not addressed at all by Dedekind (Hecke proved that $\zeta_K(s)$ extends to a meromorphic function on the complex plane in 1917). Dedekind proved that $\zeta_K(s)/\zeta(s)$ for pure cubic extensions may be written as a linear combination of Epstein zeta functions. Epstein proved in 1903 that these are meromorphic with simple poles in $s = 1$, and Artin realized in 1923 that this implies that the quotient is entire - I guess this explains his remark.

In any case Dedekind had nothing to do with this conjecture.

Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had written in 1871/72. The conjecture was named after Dedekind by van der Waall in 1974.

Despite Artin's claim, the integrality of the quotient $\zeta_K(s)/\zeta(s)$ is not addressed at all by Dedekind (Hecke proved that $\zeta_K(s)$ extends to a meromorphic function on the complex plane in 1917). Dedekind proved that $\zeta_K(s)/\zeta(s)$ for pure cubic extensions may be written as a linear combination of Epstein zeta functions. Epstein proved in 1903 that these are meromorphic with simple poles in $s = 1$, and Artin realized in 1923 that this implies that the quotient is entire - I guess this explains his remark.

In any case Dedekind had nothing to do with this conjecture.

Edit. Landau [Über die Wurzeln der Zetafunktion eines algebraischen Zahlkörpers, Math. Ann. 79 (1919), 388-401] writes on p. 390: "I have to be careful with its formulation since it is not known whether $\zeta_k(s)/\zeta(s)$ is always an entire function".

Source Link
Franz Lemmermeyer
  • 32.6k
  • 4
  • 110
  • 215

Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had written in 1871/72. The conjecture was named after Dedekind by van der Waall in 1974.

Despite Artin's claim, the integrality of the quotient $\zeta_K(s)/\zeta(s)$ is not addressed at all by Dedekind (Hecke proved that $\zeta_K(s)$ extends to a meromorphic function on the complex plane in 1917). Dedekind proved that $\zeta_K(s)/\zeta(s)$ for pure cubic extensions may be written as a linear combination of Epstein zeta functions. Epstein proved in 1903 that these are meromorphic with simple poles in $s = 1$, and Artin realized in 1923 that this implies that the quotient is entire - I guess this explains his remark.

In any case Dedekind had nothing to do with this conjecture.