On the regularized products of some Dirichlet series

Let $0<b_{1}\leq b_{2}\leq\ldots$ be a sequence of real numbers, and assume that

converges absolutely on $\mathrm{Re}(s)\gg 0$, and can be continued meromorphically to the whole complex plane and holomorphic at $s=0$. The regularized determinant of $b_{1},b_{2},\ldots$ is defined as follows

When $(b_{k})_{k=1,2,\ldots}$ is the spectrum of a positive self-adjoint operator $A$, then using techniques of heat-kernels, one can show that $\zeta_{A}(s)=\sum_{k=1}^{\infty}b_{k}^{-s}$ can be continued analytically to $s=0$, and $\operatorname*{\mathchoice{\ooalign{$\displaystyle\prod$\cr$\displaystyle\coprod$\cr}}{\ooalign{$\textstyle\prod$\cr$\textstyle\coprod$\cr}}{\ooalign{$\scriptstyle\prod$\cr$\scriptstyle\coprod$\cr}}{\ooalign{$\scriptscriptstyle\prod$\cr$\scriptscriptstyle\coprod$\cr}}}_{k=1}^{\infty}b_{k}$ is called the regularized determinant of the operator $A$ (see [16]).

The zeta regularization method can be applied to the sequence $(n+x)_{n\in\mathbb{N}}$ where $x>0$ (Here, $\mathbb{N}=\{0,1,2,\ldots\}$). We obtain the regularized product $\operatorname*{\mathchoice{\ooalign{$\displaystyle\prod$\cr$\displaystyle\coprod$\cr}}{\ooalign{$\textstyle\prod$\cr$\textstyle\coprod$\cr}}{\ooalign{$\scriptstyle\prod$\cr$\scriptstyle\coprod$\cr}}{\ooalign{$\scriptscriptstyle\prod$\cr$\scriptscriptstyle\coprod$\cr}}}_{n=0}^{\infty}(n+x)$, which can be viewed as a function in the variable $x$, and it can be continued analytically to $\mathbb{C}$. The classical Gamma function $\Gamma(x)$ can be defined from the Hurwitz zeta function via the Lerch formula [15],

where $\zeta_{H}(s,x)=\sum_{k=0}^{\infty}(k+x)^{-s}$ for $x>0$ (the Hurwitz zeta function).

Only a few general methods for the exact evaluation of $\operatorname*{\mathchoice{\ooalign{$\displaystyle\prod$\cr$\displaystyle\coprod$\cr}}{\ooalign{$\textstyle\prod$\cr$\textstyle\coprod$\cr}}{\ooalign{$\scriptstyle\prod$\cr$\scriptstyle\coprod$\cr}}{\ooalign{$\scriptscriptstyle\prod$\cr$\scriptscriptstyle\coprod$\cr}}}_{k=1}^{\infty}b_{k}$ are available. Many authors aimed to give complete results for a number of cases where it was possible to find formulas for the zeta invariants in terms of some known special functions, see for instance [3, 4, 12, 14, 19]. In [3, 4], the authors determine the regularized product of the spectrum of the Laplacian on the $D$-dimensional ball.

In the case of the complex projective space $\mathbb{P}^{n}(\mathbb{C})$ endowed with its Fubini-Study metric, the computation of its holomorphic analytic torsion (see [16]), which is by definition a linear combination of the regularized determinants of the Laplacians $\Delta^{q}$ acting on the space of smooth $(0,q)$-forms on $\mathbb{P}^{n}(\mathbb{C})$ for $q=0,\ldots,n$ , was an important step toward the formulation of an arithmetic Riemann-Roch theorem in the context of Arakelov Geometry (see [6] for the computation of the holomorphic analytic torsion of $\mathbb{P}^{n}(\mathbb{C})$, and [17, Chapter VIII] for the general formulation of the arithmetic Riemann-Roch theorem). In [11, Section 2], we gave a new method for the computation of the regularized determinant of $\Delta^{0}$ on $\mathbb{P}^{n}(\mathbb{C})$ endowed with the Fubini-Study metric. This approach can be adapted easily to get the regularized determinant of $\Delta^{q}$ for $q=1,\ldots,n$. One can use the arithmetic Riemann-Roch theorem to compute the analytic torsion of the line bundles $\mathcal{O}(m)$, with $m\geq 0$, on $\mathbb{P}^{n}(\mathbb{C})$ endowed with their Fubini-Study metrics, this was done by L. Weng in [21]. In [8], we defined a new class of singular Laplacians $\Delta_{\overline{\mathcal{O}(m)}_{\infty}}$ associated with $\mathcal{O}(m)$ on $\mathbb{P}^{1}(\mathbb{C})$ endowed with their canonical metrics. We showed that the spectrum of $\Delta_{\overline{\mathcal{O}(m)}_{\infty}}$ can be determined explicitly in terms of Bessel functions (see [8, Theorem 1.4]). In [7], we showed that the zeta function $\zeta_{\Delta_{\overline{\mathcal{O}(m)}_{\infty}}}(s)$ associated with the spectrum of $\Delta_{\overline{\mathcal{O}(m)}_{\infty}}$ can be continued analytically to $s=0$, and using [7, Theorem 3.6] we obtained that

and, we computed the regularized determinant

(see [7, Theorem 1.1]). In [9], we developed a generalized spectral theory explaining these results.

Let $0<c_{1}\leq c_{2}\leq\ldots$ and $0<b_{1}\leq b_{2}\leq\ldots$ be two sequences of positive real numbers. We assume that $\sum_{k=1}^{\infty}b_{k}^{-s}$ and $\sum_{k=1}^{\infty}c_{k}^{-s}$ converge for $\mathrm{Re}(s)\gg 0$, and admit meromorphic continuations to $\mathbb{C}$ which are holomorphic at $s=0$. An interesting question is to evaluate the following ratio:

This question was raised in [17, p. 100]. In [10], we give a partial answer to this question. Garate and Friedman in [5] develop a different method which allows them to compute some special cases.

In this paper, we use the main result of [10] to evaluate the following regularized products

where $m$ is a positive integer and $\varepsilon^{2}=1$. Our main result is,

In this situation, we see that the regularized determinant is multiplicative, i.e. (2) is equal to $1$. In general, (2) may be $\neq 1$, see [10, Eq. 11].

Let us review some basic facts about the classical Gamma function. Let $\Gamma(z)$ be the classical Gamma function. This function can be defined as follows

Bohr and Mollerup used this definition to give a characterization for the Gamma function (see [2, p.14] for more details). We know that

The Gamma function is a meromorphic function in the complex plane, with simple poles in $\mathbb{Z}_{\leq 0}$. Weierstrass theory shows that

where $\gamma$ is the Euler-Mascheroni constant. Using the Lerch formula, which is proved in Appendix 5, our theorem recovers the result of Kurokawa and Wakayama [13]. Our approach is new and self-contained. Kurokawa and Wakayama use of the infinite product expression (6) for the Gamma function and the Lerch’s formula (1) which is a crucial step in their proof.

As an application of Theorem 1.1, we obtain the following identities:

and

for any $y\in\mathbb{R}\setminus\{0\}$ (see Theorem 4.1). From the first identity, we obtain a new proof for Euler’s theorem:

By considering the Laurent expansion near $y=0$ of the third identity, we obtain a second proof for Euler’s theorem.

Motivated by the above results, we established the following theorem which generalizes identity (7).

Note that the left hand side of (8) is Mellin’s transform of

evaluated at $s=1$. This function is studied Section 3. When $m=2$ and $x=y=0$, this is the classical Jacobi theta series

It follows from the Poisson summation formula that $\theta_{2}(t,0,0)=\tfrac{\sqrt{\pi}}{\sqrt{t}}\theta_{2}(\tfrac{1}{t},0,0)-\frac{1}{2}.$ Hence, we obtain

Let $0<b_{1}\leq b_{2}\leq\ldots$ be an unbounded and nondecreasing sequence of positive real numbers. To this data, we associate the following series

We suppose that $\theta$ satisfies the following three conditions:

1. ($\Theta$1):

   The series $\sum_{k}e^{-b_{k}t}$ converges for any $t>0$,

2. ($\Theta$2):

   The series $\theta(t)$ admits an asymptotic expansion for $t\rightarrow 0^{+}$

   $$\theta(t)\sim\sum_{m\in\mathbb{N}}c_{i_{m}}t^{i_{m}},$$

   where $(i_{n})_{n\in\mathbb{N}}$ is an unbounded and non-decreasing sequence of real numbers. Moreover, we assume that $c_{i_{0}}\neq 0$,

3. ($\Theta$3):

   $\sum_{k}e^{-b_{k}t}=O(e^{-\kappa t})$ for $t\rightarrow+\infty$, for some positive real number $\kappa$.
   

We set

and call it the index of $\theta$. The zeta function $\xi$ associated with $\theta$, is by definition the function

The properties $(\Theta 1),(\Theta 2)$ and $(\Theta 3)$ imply that $\xi$ can be continued into a meromorphic function on the whole complex plane, which is holomorphic at $s=0$.

In the sequel, we consider the following sequence $(Q(k))_{k\in\mathbb{N}}$, where $Q$ is a monic polynomial of degree $\ell$ having only distinct roots with positive real part, and $Q(k)>0$ for any $k=1,2,\ldots$ It is known that the Dirichlet series

converges on any compact subset of $\mathrm{Re}(s)>1/\ell$, and has a meromorphic continuation to the whole complex plane (see [10, Theorem 3.2]). The regularized product of the sequence $(Q(k))_{k=1,2,\ldots}$ is by definition

We set

(see Appendix 5 for the properties of this function).

We have

Using Lerch’s formula, we obtain

For $m=1,2,\ldots$, $x\geq 0$ and $y\in\mathbb{C}$, we set

The following proposition is important in our study of the variation of

as a function in the variable $y$.

We have

(see [13, p. 943]) which can be proved using the reflection formula for the Gamma function.