Analytic Continuation And Zeros Of The Two-Variable Zeta Function Ζ₂(s)

The fascinating world of zeta functions extends beyond the well-known Riemann zeta function, delving into multivariable analogues that exhibit intriguing properties. This article focuses on exploring the analytic continuation and zeros of a specific two-variable zeta function, denoted as ζ₂(s). This function is defined by a double summation, adding an extra layer of complexity compared to its single-variable counterpart. We will embark on a journey to understand its behavior beyond its initial domain of convergence, unraveling its analytic structure and seeking to identify its zeros. Understanding the zeta function, ζ₂(s), requires a deep dive into the realms of multivariable calculus and analytic continuation. Our exploration will involve maneuvering through complex planes and carefully analyzing the convergence of infinite series. This quest is not merely an academic exercise; it holds potential implications for various fields, including number theory and mathematical physics. The ability to analytically continue such functions and locate their zeros can provide profound insights into the distribution of prime numbers and the behavior of dynamical systems. We aim to present a comprehensive analysis that is both mathematically rigorous and accessible to a broad audience of enthusiasts and researchers.

Defining the Two-Variable Zeta Function ζ₂(s)

Let us formally define the two-variable zeta function under investigation. For a complex variable s, we define ζ₂(s) as follows:

$$\zeta_2(s) = \sum_{n>0} \sum_{m>0} \dfrac{1}{m n^s + n m^s}$$

This double summation represents an infinite sum over all pairs of positive integers (m, n). The summand involves a combination of terms with n raised to the power of s and m raised to the power of s, adding an interesting symmetry to the function. The initial domain of convergence for this series is crucial to establish. By analyzing the asymptotic behavior of the summand, one can show that the series converges absolutely for complex values of s with a real part greater than 2, i.e., Re(s) > 2. This condition ensures that the terms in the summation decay sufficiently rapidly to guarantee convergence. The convergence analysis often involves techniques from real and complex analysis, such as the comparison test and the ratio test. Beyond this region of convergence, the function is not defined by the series representation, necessitating the use of analytic continuation techniques. These techniques allow us to extend the domain of definition of ζ₂(s) to a larger region in the complex plane, revealing its hidden structure and behavior. The process of analytic continuation is a powerful tool in complex analysis, allowing us to define functions beyond their initial domain of convergence. It relies on the uniqueness of analytic functions, ensuring that any extension obtained is the only possible analytic extension. In the case of ζ₂(s), analytic continuation may involve the use of integral representations or other functional equations that hold in a larger domain.

Convergence Analysis for Real s > 2

As initially stated, the double summation defining ζ₂(s) converges for real values of s greater than 2. To rigorously demonstrate this, we can employ a comparison test. First, observe that for positive integers m and n, we have:

$$m n^s + n m^s > m m^s + n n^s$$

Without loss of generality, assume m >= n, then we have

$$m n^s + n m^s > n n^s + n n^s = 2n^{s+1}$$

Thus,

$$\dfrac{1}{m n^s + n m^s} < \dfrac{1}{2n^{s+1}}$$

Now, we can compare the double summation with a simpler series. Since the series converges for s > 2, the original double summation also converges for s > 2. This comparison provides a solid foundation for understanding the function's behavior in the region of convergence. The rate of convergence is also an important consideration. The faster the series converges, the easier it is to numerically approximate the function's values. In this case, the convergence is relatively slow, requiring careful numerical techniques for accurate evaluation. Further analysis can reveal the nature of the convergence, such as absolute or conditional convergence. Absolute convergence is generally easier to work with, as it allows for rearranging the terms in the summation without affecting the result. The convergence analysis sets the stage for exploring the analytic continuation of ζ₂(s). Once we have established the region of convergence, we can begin to investigate methods for extending the function's definition beyond this region. This often involves finding an alternative representation of the function, such as an integral representation, that is valid in a larger domain.

The Challenge of Analytic Continuation

Analytic continuation is a pivotal concept in complex analysis, allowing us to extend the domain of definition of a function beyond its initial region of convergence. However, for functions as intricate as ζ₂(s), the process can be quite challenging. The direct summation representation is inadequate for values of s outside the region Re(s) > 2, necessitating the exploration of alternative representations. One common approach involves seeking an integral representation of the function. Integral representations often provide a way to define a function in a larger domain, as the integral may converge even when the original series does not. Another strategy is to derive a functional equation for ζ₂(s). Functional equations relate the values of the function at different points in the complex plane, effectively providing a way to