Continuation of certain Dirichlet series. (Prolongement de certaines séries de Dirichlet.) (French) Zbl 0553.10033
Shintani has shown that for a totally real number field its Dedekind zeta function has analytic continuation to the whole complex plane and its value at negative integers are rationals, by expressing the zeta function as a finite sum of the type \(\sum_{m\in {\mathbb{N}}^r}P(m+a)^{-s}\) where \(P(x)\in\mathbb{Q}(X_1,X_2,\dots,X_r)\) is a product of homogeneous polynomials of the first degree.
In this paper, the author considers analytic continuation of two kinds of Dirichlet series
\[ Z(P,b)(s)=\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}m^{b_1- 1} n^{b_2-1} P(m,n)^{-s} \tag{1} \]
where \(b_1,b_2\) are positive rational numbers and \(P(X,Y)=X^{m_1}+Y^{m_2}+X^{\alpha_ 1}Y^{\alpha_2}\), with \((m_1,m_2)\in \mathbb{Z}^2\), \(m_1,m_2\neq 0\) and \((\alpha_1,\alpha_2)\in\mathbb{N}^2\).
\[ Z(P,b,a)(s)=\sum_{m\in\mathbb{N}^r} (m_1+a_1)^{b_1}\cdots (m_r+a_r)^{b_r} P(m_1+a_1,\dots,m_r+a_r), \tag{2} \]
where \(b_1,\dots,b_r\) are positive integers and \(a_1,\dots, a_r\) are positive real numbers, and \[ P(X_1,\dots,X_r)=X_1^{m_1}+\dots+X_r^{m_r}+R(X_1,\dots,X_r),\] with \[ R(X_1,\dots,X_r)=\sum C_{(\alpha_1,\dots,\alpha_r)}X_1^{\alpha_1} X_2^{\alpha_2} \cdots X_r^{\alpha_r},\quad \sum^{r}_{i=1}\alpha_i/m_i\leq 1.\]
For the first kind, the author obtains its analytic continuation to the whole complex plane and computes residues at poles. But the situation is different according to d, where \(d=\alpha_1m_2+\alpha_2m_1-m_1m_2.\) If \(d\leq 0\), then \(Z(P,b)(s)\) has only simple poles at some rationals which are not negative integers. If \(d>0\) and if there exists a couple of positive integers \((v_1,v_2)\) such that \((b_1+m_1v_ 1)/\alpha_1=(b_2+m_2v_2)/\alpha_2,\) then it has double poles and has simple poles at negative integers. Also, if \(d\leq 0\), residues at poles and values at negative integers are expressed by some kind of integrals and Bernoulli numbers, whereas, for \(d>0\), they are expressed by values of Riemann’s zeta-function at rationals and positive integers.
For the second kind, the author also obtains its analytic continuation to the whole complex plane whose poles are simple, and residues at poles. Also the author shows that in certain cases values of \(Z(P,b,a)(s)\) at negative integers are rationals.
In this paper, the author considers analytic continuation of two kinds of Dirichlet series
\[ Z(P,b)(s)=\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}m^{b_1- 1} n^{b_2-1} P(m,n)^{-s} \tag{1} \]
where \(b_1,b_2\) are positive rational numbers and \(P(X,Y)=X^{m_1}+Y^{m_2}+X^{\alpha_ 1}Y^{\alpha_2}\), with \((m_1,m_2)\in \mathbb{Z}^2\), \(m_1,m_2\neq 0\) and \((\alpha_1,\alpha_2)\in\mathbb{N}^2\).
\[ Z(P,b,a)(s)=\sum_{m\in\mathbb{N}^r} (m_1+a_1)^{b_1}\cdots (m_r+a_r)^{b_r} P(m_1+a_1,\dots,m_r+a_r), \tag{2} \]
where \(b_1,\dots,b_r\) are positive integers and \(a_1,\dots, a_r\) are positive real numbers, and \[ P(X_1,\dots,X_r)=X_1^{m_1}+\dots+X_r^{m_r}+R(X_1,\dots,X_r),\] with \[ R(X_1,\dots,X_r)=\sum C_{(\alpha_1,\dots,\alpha_r)}X_1^{\alpha_1} X_2^{\alpha_2} \cdots X_r^{\alpha_r},\quad \sum^{r}_{i=1}\alpha_i/m_i\leq 1.\]
For the first kind, the author obtains its analytic continuation to the whole complex plane and computes residues at poles. But the situation is different according to d, where \(d=\alpha_1m_2+\alpha_2m_1-m_1m_2.\) If \(d\leq 0\), then \(Z(P,b)(s)\) has only simple poles at some rationals which are not negative integers. If \(d>0\) and if there exists a couple of positive integers \((v_1,v_2)\) such that \((b_1+m_1v_ 1)/\alpha_1=(b_2+m_2v_2)/\alpha_2,\) then it has double poles and has simple poles at negative integers. Also, if \(d\leq 0\), residues at poles and values at negative integers are expressed by some kind of integrals and Bernoulli numbers, whereas, for \(d>0\), they are expressed by values of Riemann’s zeta-function at rationals and positive integers.
For the second kind, the author also obtains its analytic continuation to the whole complex plane whose poles are simple, and residues at poles. Also the author shows that in certain cases values of \(Z(P,b,a)(s)\) at negative integers are rationals.
Reviewer: K. I. Ohta (Tokyo)
MSC:
| 11M41 | Other Dirichlet series and zeta functions |
| 30B50 | Dirichlet series, exponential series and other series in one complex variable |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
