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On a factorization of Riemann’s ζ function with respect to a quadratic field and its computation
- Source :
- Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas; September 2012, Vol. 106 Issue: 2 p419-427, 9p
- Publication Year :
- 2012
-
Abstract
- Abstract: Let K be a quadratic field, and let ζ <subscript> K </subscript> its Dedekind zeta function. In this paper we introduce a factorization of ζ <subscript> K </subscript> into two functions, L <subscript>1</subscript> and L <subscript>2</subscript>, defined as partial Euler products of ζ <subscript> K </subscript>, which lead to a factorization of Riemann’s ζ function into two functions, p <subscript>1</subscript> and p <subscript>2</subscript>. We prove that these functions satisfy a functional equation which has a unique solution, and we give series of very fast convergence to them. Moreover, when Δ<subscript> K </subscript> > 0 the general term of these series at even positive integers is calculated explicitly in terms of generalized Bernoulli numbers.
Details
- Language :
- English
- ISSN :
- 15787303 and 15791505
- Volume :
- 106
- Issue :
- 2
- Database :
- Supplemental Index
- Journal :
- Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
- Publication Type :
- Periodical
- Accession number :
- ejs26826776
- Full Text :
- https://doi.org/10.1007/s13398-012-0060-z