交错Dirichlet Lambda函数能否用zeta函数或其他Dirichlet L函数表示？

Hey there! Great question about that alternating series—it's actually a well-known function in number theory called the Dirichlet lambda function, often written as $\lambda(s)$. Your series is exactly its definition:

$$\lambda(s) = \sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$$

Let's break down its connections to the Riemann zeta function, other Dirichlet L-functions, and key properties that might help you with its values beyond $s=1$:

1. It's a Dirichlet L-Function #

First off, your series is equivalent to the Dirichlet L-function for the non-principal character modulo 4 (denoted $\chi_4$). This character is defined as:

• $\chi_4(n) = 1$ if $n \equiv 1 \pmod{4}$
• $\chi_4(n) = -1$ if $n \equiv 3 \pmod{4}$
• $\chi_4(n) = 0$ if $n$ is even

When you write the L-function for $\chi_4$, you get exactly your series:
$$L(s, \chi_4) = \sum_{n=1}^\infty \chi_4(n) n^{-s} = \sum_{k=0}^\infty (4k+1)^{-s} - \sum_{k=0}^\infty (4k+3)^{-s} = \lambda(s)$$

2. Connection to the Riemann Zeta Function #

You can express $\lambda(s)$ in terms of the Riemann zeta function $\zeta(s)$ using the Hurwitz zeta function $\zeta(s, a)$ (a generalization of the Riemann zeta that starts summing from an offset $a$):
$$\lambda(s) = 4^{-s} \left( \zeta\left(s, \frac{1}{4}\right) - \zeta\left(s, \frac{3}{4}\right) \right)$$

For context, the Hurwitz zeta function is defined as $\zeta(s, a) = \sum_{k=0}^\infty (k+a)^{-s}$, so this directly links your alternating series to the standard zeta function's structure. You can also derive a supporting relation by splitting $\zeta(s)$ into even and odd terms:

• The Riemann zeta function can be written as $\zeta(s) = \sum_{\text{even }n} n^{-s} + \sum_{\text{odd }n} n^{-s} = 2^{-s}\zeta(s) + \sum_{\text{odd }n} n^{-s}$
• Rearranging gives $\sum_{\text{odd }n} n^{-s} = (1 - 2^{-s})\zeta(s)$

While this isn't your alternating series, it's a key building block for understanding how $\lambda(s)$ fits into the zeta function's framework.

3. Special Values Beyond $s=1$ #

You already know $\lambda(1) = \frac{\pi}{4}$ (Leibniz's formula). Here are some other useful special values:

• Even positive integers: For $s=2k$ where $k$ is a positive integer, $\lambda(s)$ has closed-form expressions in terms of $\pi$ and Euler numbers. Examples include:
  • $\lambda(2) = \frac{\pi^2}{8}$
  • $\lambda(4) = \frac{\pi^4}{96}$
  • $\lambda(6) = \frac{\pi^6}{960}$
• Odd integers greater than 1: These values are irrational but don't have simple $\pi$-based closed forms. They're part of a class of numbers called Dirichlet L-values, which are deeply studied in number theory (often connected to modular forms and arithmetic geometry).

4. Analytic Continuation & Functional Equation #

Like the Riemann zeta function, $\lambda(s)$ can be analytically continued to the entire complex plane—unlike $\zeta(s)$, though, it has no poles (it's an entire function). It also satisfies a functional equation that relates its values at $s$ and $1-s$:
$$L(1 - s, \chi_4) = \frac{2}{\pi^s} \Gamma(s) \cos\left(\frac{\pi s}{2}\right) L(s, \chi_4)$$
Here, $\Gamma(s)$ is the gamma function, which extends the factorial function to complex numbers.

Hopefully this clears up the connections you were looking for! The core idea is that your series is a fundamental Dirichlet L-function with tight links to the Riemann zeta function and well-documented special values.

内容的提问来源于stack exchange，提问作者aleden