# Odlyzko bounds

Effective lower bounds for , the minimal value of the discriminant of algebraic number fields having signature (i.e. having real and non-real conjugates), obtained in 1976 by A.M. Odlyzko. See also Algebraic number; Number field.

The first such bound was proved in 1891 by H. Minkowski [a4], who showed

(a1) |

with . He obtained it using methods from the geometry of numbers; the same method was used later by several authors to improve (a1) (see [a5] for the strongest result obtained in this way).

In 1974, H.M. Stark ([a11], [a12]) observed that Hadamard factorization of the Dedekind zeta-function leads to a formula expressing by the zeros of and the value of its logarithmic derivative at a complex number with . He used this formula with a proper choice of to deduce lower bounds for which were essentially stronger than Minkowski's bound, but did not reach the bounds obtained by geometrical methods.

In 1976, Odlyzko [a7] (cf. [a9]) modified Stark's formula and obtained the following important improvement of (a1):

(a2) |

with .

In particular, one has

If the extended Riemann hypothesis is assumed (cf. also Riemann hypotheses; Zeta-function), then the constants and in (a2) can be replaced by and , respectively. For small degrees the bound (a2) can be improved (see [a2], [a10]) and several exact values of are known.

On the other hand, it has been shown in [a13], as a consequence of their solution of the class field tower problem (cf. also Tower of fields; Class field theory), that is finite. The best explicit upper bound for it, , is due to J. Martinet [a1], who obtained this as a corollary of his constructions of infinite -class towers of suitable fields.

For surveys of this topic, see [a9], [a3] and [a8].

#### References

[a1] | J. Martinet, "Tours de corps de classes et estimations de discriminants" Invent. Math. , 44 (1978) pp. 65–73 |

[a2] | J. Martinet, "Petits discriminants" Ann. Inst. Fourier (Grenoble) , 29 : fasc.1 (1979) pp. 159–170 |

[a3] | J. Martinet, "Petits discriminants des corps de nombres" , Journ. Arithm. 1980 , Cambridge Univ. Press (1982) pp. 151–193 |

[a4] | H. Minkowski, "Théorèmes arithmétiques" C.R. Acad. Sci. Paris , 112 (1891) pp. 209–212 (Gesammelte Abh. I (1911), 261-263, Leipzig–Berlin) |

[a5] | H.P. Mulholland, "On the product of complex homogeneous linear forms" J. London Math. Soc. , 35 (1960) pp. 241–250 |

[a6] | A. Odlyzko, "Some analytic estimates of class numbers and discriminants" Invent. Math. , 29 (1975) pp. 275–286 |

[a7] | A. Odlyzko, "Lower bounds for discriminants of number fields" Acta Arith. , 29 (1976) pp. 275–297 (II: Tôhoku Math. J., 29 (1977), 275-286) |

[a8] | A. Odlyzko, "Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results" Sém. de Théorie des Nombres, Bordeaux , 2 (1990) pp. 119–141 |

[a9] | G. Poitou, "Minoration de discriminants (d'aprés A.M. Odlyzko)" , Sém. Bourbaki (1975/76) , Lecture Notes in Mathematics , 567 , Springer (1977) pp. 136–153 |

[a10] | G. Poitou, "Sur les petits discriminants" Sém. Delange–Pisot–Poitou , 18 : 6 (1976/77) |

[a11] | H.M. Stark, "Some effective cases of the Brauer–Siegel theorem" Invent. Math. , 23 (1974) pp. 135–152 |

[a12] | H.M. Stark, "The analytic theory of numbers" Bull. Amer. Math. Soc. , 81 (1975) pp. 961–972, |

[a13] | E.S. Golod, I.R. Shafarevich, "On the class-field tower" Izv. Akad. Nauk. SSSR , 28 (1964) pp. 261–272 (In Russian) |

**How to Cite This Entry:**

Odlyzko bounds. WÅ‚adysÅ‚aw Narkiewicz (originator),

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