An Upper Bound for Hecke Zeta-Functions with Groessencharacters
✍ Scribed by Peter Söhne
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 422 KB
- Volume
- 66
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
An estimate for Hecke Zeta-functions with Gro ssencharacters on the critical line is proved which corresponds to the classical result `(1Â2+it)R = (|t| +1) 1Â6+= on Riemann's zeta-function. The constants implied in the R-sign depend neither on the conductor nor on the exponents of the Gro ssencharacter.
1997 Academic Press Kaufman [4] and were able to estimate the Dedekind Zeta-function of an algebraic number field K of degree n over the rationals in the form
Heath-Brown applied an n-dimensional variant of van der Corput's method. In this paper we follow that method to give a similar estimate for zeta-functions of K, whose coefficients are Gro ssencharacters in the sense of Hecke [3].
The subject matter of this paper forms part of the authors thesis (Marburg, 1990). He thanks Professor Dr. W. Schaal and Professor Dr. J. Hinz for fruitful discussions during the preparation of it.
For the conjugates of any : # K let : (q) # R (1 q r 1 ), : (q+r2) =: (q) (r 1 +1 q r 1 +r 2 =r+1).
Then a ``Gro ssencharacter for numbers'' to the modulus f, f an integral ideal of K, is given by
q=r 1 +1 \ : (q) |: (q) |+ 2aq |: (q) | &2ibq ;
( 2 ) article no. NT972167 225 0022-314XÂ97 25.00