数理解析研究所講究録
1091 巻 1999 年 172-174

172

Some curious Dirichlet series
Shigeki Egami
Faculty of Engneering,$\perp \mathrm{o}\mathrm{y}\mathrm{a}\mathrm{m}\mathrm{a}$unlverslTy

3190 Gofuku, Toyama city Toyama 930, Japan
megami@eng.toyama-u.ac.jp
Problems on analytic continuation of ”number theoretical” Dirichlet series
constitute an important part of analytic number theory. There are some wellknown methods such as Poisson summmation formula, Euler-Maclaurin formula,contour integration, etc.. In this paper we discuss some marginal Dirichlet series
for which the above methods seems to be useless.

1Fibonacci zeta
be the Fibonacci sequence defined by $F_{1}=F_{2}=1,$ $p_{n+2}=F_{n+}1+$. DefineLet$\{F_{n}\}_{n\in N}$$F_{n},$$(n\geq 3)$$Fib \langle s\rangle=n1\sum_{=}\frac{1}{F_{n}^{S}}\infty$.

$\langle 1\rangle$

It is easily seen that the above series converges absolutely and uniformly in thewide sence in the region $\Re s>0$ . Recently the problem of transcendency of Fib $(s)$at positive integers became reachable by the advances of the theory of Mahler
functions.

Proposition 1 (D. Deverney, Ke. Nishioka, Ku. Nishioka, I. Shiokawa)
are $tra,n,9Cend,entd$ .
For $e?;e,n$ positive integers$s,$$F\dot{i}b(.9)$

are c,onjectured but
The corresponding results for odd integersunsolYedtll]). The results have a curious similarity to the corresponding ones fortheinterest to. So, it is \‘oftheproperties of Fib $(s)$ as function of variable . We have$(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}1)$$\overline{\mathrm{c}}\mathrm{l}\mathrm{i}\mathrm{s}\overline{\mathrm{C}}\mathrm{u}\mathrm{S}\overline{\mathrm{S}}$$\mathrm{R}\mathrm{i}\overline{\mathrm{e}}\overline{\mathrm{I}\mathrm{u}}\mathrm{a}\mathrm{n}_{\mathrm{I}1}\overline{\prime\Delta}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{f}\iota\ln\overline{(}j\mathrm{t}\mathrm{i}\mathrm{c}\overline{\mathrm{J}}\mathrm{I}^{-}1\mathrm{S}$$\overline{\mathrm{a}}\mathrm{I}^{-}1\mathrm{a}1\overline{\mathrm{y}}\mathrm{t}\mathrm{i}\mathrm{c}$$\overline{\mathrm{s}}’ 0\overline{\mathrm{I}\mathrm{I}}\mathrm{l}\mathrm{e}$$s$Theorem 1 Fib $(s)$ can be continued to a meromorphic function holomorphic
where is a non-negative integer, in
exept the simple poles at.
an integer, and$s=-2k+ \frac{\pi in}{\log\alpha}f$$\alpha=\frac{1+\sqrt{5}}{2}$

$k$

$n$

173

The proof is quite easy. Rom the well known expression of Fibonacci numbers:(2)$F_{n}= \frac{\alpha^{n}-(-\alpha)^{-n}}{\sqrt{5}}$we have$F_{\dot{i}}b(s)$$=$$\sqrt{5}^{s}\sum_{\hslash=0}^{\infty}\frac{1}{(\alpha^{n}-(-\alpha)^{-n})s}$$=$$\sqrt{5}^{s}(\sum_{m=1}^{\infty}\frac{1}{(\alpha^{2m}+\alpha^{-2m})^{s}}+\sum_{m=0}^{\infty}\frac{1}{(\alpha^{2m+1}-\alpha-2m-1)^{S}})$$=$$\sqrt{5}^{s}(\sum_{m=1}^{\infty}\frac{1}{\alpha^{2ms}}(1+\alpha^{-4m})^{s}+\sum_{m=0}\frac{1}{\alpha^{(2m+1)_{S}}}\infty(\iota-\alpha-(4m+2))\delta)$$=$$\sqrt{5}^{s}(g(s)+f(s))$(set)

From the binary expansion and the sum formula of geometric series we have$g(s)=k=0 \sum\infty(_{k}^{-}S)\frac{1}{\alpha^{2\mathit{8}+4}-k1}$(3)$f(s)= \sum_{k=0}^{\infty}(^{-}k\delta)\frac{\alpha^{-s-2k}}{1-\alpha^{-2\mathit{8}-4k}}$

(4)

and
which give meromorphic continu\’ation.

2

Additive convolution

Let
(5)

$f_{1}(s)=n= \sum\frac{a_{1}(n)}{n^{s}}\infty 1’\ldots,$ $f_{r}(S)=n1 \sum_{=}\frac{a_{r}(n)}{n^{s}}\infty$be Dirichlet series with ”nice” properties. What properties are inherited by their”additive convolution” i.e.
$f_{1} \theta\cdot\cdot \mathrm{r}\circ f_{r}(s)=\sum_{n_{1},\ldots,n_{f}=1}\frac{a_{1}(n_{1}).\cdot.\cdot.\cdot a_{r}(n)r}{(n_{1}++n_{r})^{s}}\infty$.

(6)

Note that, when are the Riemann zeta function the additive convolution become -ple zeta function of Barnes. This example suggest the analytic continuability ofin general. In fact we can prove$f_{i}$$\mathrm{r}$$f_{1}\mathrm{O}\cdots\theta f’(s)$$(i=1, \ldots, r)$ can be continued to meromorphic
Theorem 2 Iffunctions
holomorphic exept finite number of poles and of polynomial growth dong the imaginary direction, then so is their additive convolution$f_{i}(s),$$f_{1}\Leftrightarrow\cdots\Leftrightarrow fr(S)$174

The proof is based on the following integral expression:

Lemma 1$\sum_{n_{1},\ldots,n_{r}=1}^{\infty}\frac{a_{1}(n_{1}).\cdots a_{r}(n)r}{(n_{1}+\cdot\cdot+n_{r}+1)s}=$$\frac{1}{(2\pi\dot{i})^{r}}\int_{(c_{1})}$

where$c_{i}$,$\int_{(c_{\gamma})}\frac{\Gamma(s_{1})\cdots \mathrm{r}(S_{r})\Gamma(s-S_{1}-\cdots-s_{r})}{\Gamma(s)}f_{1}(S_{1})\cdots f_{r}(_{S_{r}})ds1\ldots ds_{r}$is a real number in the absolute convergent region of$f_{i}(s)$.

Note that the left hand side is not exactly the additive convolution. But thedifference can be easily recovered by binary expansion of the denominator. The
analytic continuation is done by shift of the lines of integration. The detail ofthe proof will appear elsewhere.
The same method can be applied successfully even more general Dirichlet
series. We discus(7),$G(s)=n1 \sum_{=}\frac{g(n)}{n^{s}}\infty$

where $g(n)= \sum_{m+k=n}\Lambda(m)\Lambda(k)$ , which is interesting from the viewpoint of additive prime theory(cf. [2]). Though this case does not satisfy the assumptionsof Theorem 2 we have

Theorem 3 On the assumption of the Riemann hypothesis, $G(s)$ can be continued to a function meromomrphic in the region $\Re s>1$ and have expression$a$$c(s)= \frac{2}{(s-2)(S-1)}-\frac{1}{\Gamma(s)}\sum_{\rho}\Gamma(\rho)\mathrm{r}(_{S-1-\rho})+J(S)$,zeros of the Riemann zeta
where the summation runs over all the$\Re
s>1$, and $J(s)=O((\Im_{S))}1+\epsilon$ forfunction, $J(s)\dot{u}$ a function holomorphic in
any in this region.
$nonarrow tri\dot{m}al$$\epsilon$

References[1] D. Duverney, Ke. Nishioka, Ku. Nishioka, I. shiokawa, Transcendence ofRoger-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, RIMS Proceeding, 1060(1998),91-100[2] A. Fujii, Acta Arith. 1993?