I am not sure whether I understand your question, so let me know when the following is not what you asked for. 

 - E. Landau, 
   <em> Verallgemeinerung eines {\it Polya}schen Satzes auf algebraische Zahlk&ouml;rper</em>
    G&ouml;tt. Nachr. 1918, 478--488

proved that the regulator R of a number field of degree n and discriminant D satisfies
$R < c(n) \cdot \sqrt{D} \cdot \log^{n-1} D$. Landau's methods were improved upon by

 - R. Remak, 
   <em> Elementare Absch&auml;tzungen von Fundamentaleinheiten und des Regulators eines 
        algebraischen Zahlk&ouml;rpers</em>,
  J. Reine Angew. Math. 167 (1932), 360-378.