×

On analytic continuation and functional equation of certain Dirichlet series. (English) Zbl 0782.30005

The well-known functional equation of the Riemann zeta function \(\zeta(s)\) is \[ 2^{1-s} \Gamma(s) \zeta(s)\cos \left( {{s\pi} \over 2} \right)= \pi^ s \zeta(1-s),\tag{1} \] where \(s\) is a complex variable. If we set \(\xi(s)=\pi^{-{s\over 2}} \Gamma({s\over 2}) \zeta(s)\), then (1) can be written as (2) \(\xi(s)=\xi(1-s)\).
In this paper the authors investigate analytic continuation and functional equation of Riemann’s type (see the above (1) and (2)) of Dirichlet series by using one of Riemann’s methods [see E. C. Titchmarsh, The theory of the Riemann zeta function (1986; Zbl 0601.10026)].

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
30B40 Analytic continuation of functions of one complex variable

Citations:

Zbl 0601.10026
PDFBibTeX XMLCite
Full Text: DOI EuDML