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Effective lower bounds for $M ( r _ { 1 } , r _ { 2 } )$, the minimal value of the [[Discriminant|discriminant]] $| d ( K ) |$ of algebraic number fields $K$ having signature $( r _ { 1 } , r _ { 2 } )$ (i.e. having $r_1$ real and $2r_2$ non-real conjugates), obtained in 1976 by A.M. Odlyzko. See also [[Algebraic number|Algebraic number]]; [[Number field|Number field]].

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

\begin{equation} \tag{a1} M ( r _ { 1 } , r _ { 2 } ) &gt; \left( \frac { \pi } { 4 } \right) ^ { 2 r _ { 2 } } \left( \frac { n ^ { n } } { n ! } \right) ^ { 2 }, \end{equation}

with $n = r _ { 1 } + 2 r _ { 2 }$. He obtained it using methods from the [[Geometry of numbers|geometry of numbers]]; the same method was used later by several authors to improve (a1) (see [[#References|[a5]]] for the strongest result obtained in this way).

In 1974, H.M. Stark ([[#References|[a11]]], [[#References|[a12]]]) observed that Hadamard factorization of the [[Dedekind zeta-function|Dedekind zeta-function]] $\zeta _ { K } ( s )$ leads to a formula expressing $\operatorname{log} | d ( K ) |$ by the zeros of $\zeta _ { K } ( s )$ and the value of its logarithmic derivative at a complex number $s _ { 0 } \neq 0,1$ with $\zeta_{ K } ( s _ { 0 } ) \neq 0$. He used this formula with a proper choice of $s_0$ to deduce lower bounds for $M ( r _ { 1 } , r _ { 2 } )$ which were essentially stronger than Minkowski's bound, but did not reach the bounds obtained by geometrical methods.

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

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002016.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a2)</td></tr></table>

with $\operatorname { lim } _ { x \rightarrow \infty } \epsilon ( n ) = 0$.

In particular, one has

\begin{equation*} D = \liminf  _ { n \rightarrow \infty } M ( r _ { 1 } , r _ { 2 } ) ^ { 1 / n } \geq 22. \end{equation*}

If the extended Riemann hypothesis is assumed (cf. also [[Riemann hypotheses|Riemann hypotheses]]; [[Zeta-function|Zeta-function]]), then the constants $60$ and $22$ in (a2) can be replaced by $180$ and $41$, respectively. For small degrees the bound (a2) can be improved (see [[#References|[a2]]], [[#References|[a10]]]) and several exact values of $M ( r _ { 1 } , r _ { 2 } )$ are known.

On the other hand, it has been shown in [[#References|[a13]]], as a consequence of their solution of the class field tower problem (cf. also [[Tower of fields|Tower of fields]]; [[Class field theory|Class field theory]]), that $D$ is finite. The best explicit upper bound for it, $D \leq 92.4$, is due to J. Martinet [[#References|[a1]]], who obtained this as a corollary of his constructions of infinite $2$-class towers of suitable fields.

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

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  J. Martinet,   "Tours de corps de classes et estimations de discriminants"  ''Invent. Math.'' , '''44'''  (1978)  pp. 65–73</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Martinet,   "Petits discriminants"  ''Ann. Inst. Fourier (Grenoble)'' , '''29''' :  fasc.1  (1979)  pp. 159–170</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Martinet,   "Petits discriminants des corps de nombres" , ''Journ. Arithm. 1980'' , Cambridge Univ. Press  (1982)  pp. 151–193</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H. Minkowski,   "Théorèmes arithmétiques"  ''C.R. Acad. Sci. Paris'' , '''112'''  (1891)  pp. 209–212  (Gesammelte Abh. I (1911), 261-263, Leipzig–Berlin)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  H.P. Mulholland,   "On the product of $n$ complex homogeneous linear forms"  ''J. London Math. Soc.'' , '''35'''  (1960)  pp. 241–250</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Odlyzko,   "Some analytic estimates of class numbers and discriminants"  ''Invent. Math.'' , '''29'''  (1975)  pp. 275–286</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A. Odlyzko,   "Lower bounds for discriminants of number fields"  ''Acta Arith.'' , '''29'''  (1976)  pp. 275–297  (II: Tôhoku Math. J., 29 (1977), 275-286)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  G. Poitou,   "Minoration de discriminants (d'aprés A.M. Odlyzko)" , ''Sém. Bourbaki (1975/76)'' , ''Lecture Notes in Mathematics'' , '''567''' , Springer  (1977)  pp. 136–153</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  G. Poitou,   "Sur les petits discriminants"  ''Sém. Delange–Pisot–Poitou'' , '''18''' :  6  (1976/77)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  H.M. Stark,   "Some effective cases of the Brauer–Siegel theorem"  ''Invent. Math.'' , '''23'''  (1974)  pp. 135–152</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  H.M. Stark,   "The analytic theory of numbers"  ''Bull. Amer. Math. Soc.'' , '''81'''  (1975)  pp. 961–972,</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  E.S. Golod,   I.R. Shafarevich,   "On the class-field tower"  ''Izv. Akad. Nauk. SSSR'' , '''28'''  (1964)  pp. 261–272  (In Russian)</td></tr></table>