Integral Representations of Three Novel Multiple Zeta Functions for Barnes Type: A Probabilistic Approach

1. Problem Statement

The paper addresses the analytic continuation and integral representation of multiple zeta functions, specifically focusing on generalizations of the Barnes multiple zeta function ($\zeta_B$).

• Context: The Riemann and Hurwitz zeta functions are fundamental in analytic number theory. Their multidimensional generalizations, such as the Euler–Zagier and Barnes multiple zeta functions, are well-studied.
• Limitation: The classical Barnes multiple zeta function $\zeta_B(s)$ is known to extend to a meromorphic function in the complex plane $\mathbb{C}$, possessing simple poles at $s = 1, 2, \dots, r$ (where $r$ is the dimension).
• Objective: The authors aim to introduce three novel variants of the Barnes multiple zeta function. The primary goal is to determine if these new functions can be analytically continued to entire functions (functions holomorphic everywhere in $\mathbb{C}$ with no poles), which would be a significant departure from the behavior of classical Barnes zeta functions.

2. Methodology

The authors employ a probabilistic approach, leveraging the connection between zeta functions and hyperbolic probability distributions introduced by Pitman and Yor (2003).

• Probabilistic Framework:
  • The paper utilizes three specific hyperbolic random variables: $\hat{S}_t$ (sinh), $\hat{C}_t$ (cosh), and $\hat{T}_t$ (tanh), defined by their characteristic functions involving $\sinh$, $\cosh$, and $\tanh$ functions.
  • These variables are infinitely divisible and possess finite moments of all orders.
• Integral Transform Technique:
  • The authors define a moment function $h_X(s; c) = E[(c + iX)^{-s}]$ for a random variable $X$.
  • Lemma 3 establishes that if $X$ has finite moments of all orders, this transform is an entire function in $s$.
• Derivation Strategy:
  1. Express the multiple series definitions of the new zeta functions using the Gamma function identity: $\Gamma(s)z^{-s} = \int_0^\infty x^{s-1}e^{-zx}dx$.
  2. Sum the resulting series inside the integral using binomial expansions and the specific generating functions of the hyperbolic distributions.
  3. Identify the resulting integrand as the characteristic function (or moment generating function) of a linear combination of the hyperbolic random variables.
  4. Apply Fubini's theorem to interchange the order of integration and summation, converting the series into an integral representation involving the probability density functions of the hyperbolic distributions.

3. Key Contributions and Definitions

The paper defines three new multiple zeta functions, denoted as follows:

1. Multiple Hyperbolic Sinh-Moment Function ($S_{\alpha,a}(s; b)$):

   • Defined using binomial coefficients $\binom{n_j+\alpha_j-1}{n_j}$.
   • Relation: When all $\alpha_j = 1$, this reduces exactly to the classical Barnes multiple zeta function $\zeta_B(s)$.
   • Behavior: It retains the meromorphic nature of the classical Barnes zeta function.

2. Multiple Hyperbolic Cosh-Moment Function ($C_{\alpha,a}(s; b)$):

   • Defined using binomial coefficients $\binom{-\alpha_j}{n_j}$, introducing alternating signs $(-1)^{\sum n_j}$.
   • Relation: When $\alpha_j=1$, it becomes an alternating version of the Barnes zeta function.
   • Behavior: Proven to be an entire function.

3. Multiple Hyperbolic Tanh-Moment Function ($T_{\alpha,a}(s; b)$):

   • Defined involving an expectation over uniform random variables $V_j$ (sums of i.i.d. $U(0,1)$ variables) and alternating signs.
   • Behavior: Proven to be an entire function.

4. Main Results

Theorem 1: The Sinh-Moment Function

• Integral Representation:
  $$S_{\alpha,a}(s; b) = c_r \frac{\Gamma(s-\beta)}{\Gamma(s)} \int_{\mathbb{R}} \frac{1}{(c+iy)^{s-\beta}} dF_{Y_r}(y)$$
  where $Y_r$ is a sum of independent hyperbolic sinh variables.
• Analytic Properties:
  • Extends to a meromorphic function in $\mathbb{C}$.
  • Has simple poles at $s = 1, 2, \dots, \beta$ (if $\beta$ is an integer) or at non-integer points determined by $\beta$.
  • This confirms that the classical Barnes zeta function (a special case) is meromorphic, consistent with existing literature.

Theorem 2: The Cosh-Moment Function

• Integral Representation:
  $$C_{\alpha,a}(s; b) = 2^{-\beta} \int_{\mathbb{R}} \frac{1}{(c+iz)^s} dF_{Z_r}(z)$$
  where $Z_r$ is a sum of independent hyperbolic cosh variables.
• Analytic Properties:
  • Crucial Result: Unlike the sinh case, $C_{\alpha,a}(s; b)$ extends to an entire function in the whole complex plane $\mathbb{C}$.
  • There are no poles.

Theorem 3: The Tanh-Moment Function

• Integral Representation:
  $$T_{\alpha,a}(s; b) = 2^{-\beta} \int_{\mathbb{R}} \frac{1}{(b+iw)^s} dF_{W_r}(w)$$
  where $W_r$ is a sum of independent hyperbolic tanh variables.
• Analytic Properties:
  • Crucial Result: Similar to the cosh case, $T_{\alpha,a}(s; b)$ extends to an entire function in $\mathbb{C}$.
  • There are no poles.

Corollaries

The paper derives specific corollaries for lower dimensions ($r=1, 2$) and specific parameter choices, recovering known results for the Hurwitz zeta function and the alternating Hurwitz zeta function, while providing new integral representations for them using hyperbolic densities.

5. Significance

• Novelty in Analytic Continuation: The most significant finding is that the cosh and tanh variants of the Barnes multiple zeta functions are entire functions. This contrasts sharply with the classical Barnes zeta function, which always has poles. This suggests that the introduction of alternating signs (via the cosh/tanh structure) fundamentally alters the singularity structure of the function, removing the poles entirely.
• Probabilistic Insight: The paper successfully demonstrates the power of probabilistic methods (specifically hyperbolic distributions) in analytic number theory. It provides a unified framework to derive integral representations for these complex series, linking number-theoretic functions to the moments of specific probability distributions.
• New Tools for Research: The integral representations derived (Equations 6, 12, and 20) provide new tools for studying the values, asymptotic behavior, and functional equations of these zeta functions, potentially opening new avenues for research in multiple zeta values and special functions.

In summary, the paper introduces three new classes of multiple zeta functions. While one recovers the classical meromorphic behavior of the Barnes zeta function, the other two (cosh and tanh types) are shown to be entire functions, a surprising and significant result achieved through a novel probabilistic integral representation approach.