Effective Results in Number Theory - The 'Siegel Period'

The 'Siegel Period'

Many of the principal results of analytic number theory that were proved in the period 1900-1950 were in fact ineffective. The main examples were:

• The Thue-Siegel-Roth theorem
• Siegel's theorem on integral points, from 1929
• The 1934 theorem of Hans Heilbronn and Edward Linfoot on the class number 1 problem
• The 1935 result on the Siegel zero
• The Siegel-Walfisz theorem based on the Siegel zero.

The concrete information that was left theoretically incomplete included lower bounds for class numbers (ideal class groups for some families of number fields grow); and bounds for the best rational approximations to algebraic numbers in terms of denominators. These latter could be read quite directly as results on Diophantine equations, after the work of Axel Thue. The result used for Liouville numbers in the proof is effective in the way it applies the mean value theorem: but improvements (to what is now the Thue-Siegel-Roth theorem) were not.