- Siegel zero
In
mathematics , more specifically in the field ofanalytic number theory , a Siegel zero, named afterCarl Ludwig Siegel , is a type of potentialcounterexample to thegeneralized Riemann hypothesis , on the zeroes ofDirichlet L-function .There are hypothetical values "s" of a
complex variable , very near (in a quantifiable sense) to 1, such that:"L"("s",χ) = 0
for a
Dirichlet character χ, to modulus "q" say. Important results on this type of zero of anL-function were obtained in the 1930s byCarl Ludwig Siegel , from whom they take their name (he was not the first to consider them, and they are sometimes called Landau-Siegel zeroes to acknowledge also the work ofEdmund Landau ).The possibility of a Siegel zero in analytic terms leads to an ineffective estimate
:"L"("1",χ) > "C"(ε)"q"−ε
where "C" is a function of ε for which the proof provides no explicit
lower bound (seeeffective results in number theory ).The importance of the possible Siegel zeroes is seen in all known results on the zero-free regions of L-functions: they show a kind of 'indentation' near "s" = 1, while otherwise generally resembling that for the
Riemann zeta function — that is, they are to the left of the line "Re"("s") = 1, and asymptotic to it. Because of theanalytic class number formula , data on Siegel zeroes have a direct impact on theclass number problem , of giving lower bounds for class numbers. This question goes back toC. F. Gauss . What Siegel showed was that such zeroes are of a particular type (namely, that they can occur only for χ a "real" character, which must be aJacobi symbol ); and, that for each modulus "q" there can be at most one such. This was by a 'twisting' would prove that it did not exist). In subsequent developments, however, detailed information on the Siegel zero has not shown it to be impossible. Work on the class number problem has instead been progressing by methods fromKurt Heegner 's work, fromtranscendence theory , and thenDorian Goldfeld 's work combined with theGross-Zagier theorem onHeegner point s.References
*C. L. Siegel, "Über die Klassenzahl quadratischer Zahlkörper", Acta Arithmetica 1 (1936), pages 83-86
Wikimedia Foundation. 2010.
