Riemann hypothesis, generalized

A statement about the non-trivial zeros of Dirichlet -functions (cf. Dirichlet -function), Dedekind zeta-functions (cf. Zeta-function) and several other similar functions, similar to the Riemann hypothesis (cf. Riemann hypotheses) on the non-trivial zeros of the Riemann zeta-function . In the case of Dirichlet zeta-functions the generalized Riemann hypothesis is called the extended Riemann hypothesis.

Comments

For Dirichlet -functions it is not even known whether there exist real zeros in the interval (Siegel zeros). This is important in connection with the class number of quadratic fields (see also Quadratic field; Siegel theorem).

Let be an algebraic number field, the group of fractional ideals of and its idèle class group (cf. Idèle; Fractional ideal). Let be a quasi-character on , i.e. a continuous homomorphism of into the group of non-zero complex numbers. Then for an idèle one has , where for each , is a quasi-character of which is equal to unity on , the units of the local completion , for almost-all . Let be a finite subset of the valuations on including the Archimedian ones, . A function can now be defined on by setting for all prime ideals ,

and extending multiplicatively. These functions are called Hecke characters or Grössencharakters. Given such a character, the Hecke zeta-function of is defined by

where is the absolute norm . The function is also called -series, Dirichlet -series (when is a Dirichlet character) or Hecke -function with Grössencharakter; it is also denoted by . If one obtains the Dedekind -function. For Dirichlet -series the generalized Riemann hypothesis states that if .

References

[a1]	H. Heilbronn, "Zeta-functions and -functions" J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) pp. 204–230
[a2]	W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , Springer & PWN (1990) pp. Chapt. 7, §1