Dirichlet Beta Values: Analysis & Representations

• Dirichlet beta values are derived from an alternating series over odd integers and serve as fundamental constants in analytic number theory.
• They exhibit distinct representations: odd-indexed values are expressed via Euler numbers, while even-indexed values involve polygamma functions and integral formulations.
• Studies of these values enhance our understanding of irrationality and transcendence, connecting deep arithmetic structures like Euler and Bernoulli numbers.

The Dirichlet beta values are the special values of the Dirichlet beta function $\beta(s)$, a prototypical $L$-series defined by $\beta(s)=\sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^s}$ for $\Re(s)>0$. This function is of central importance in analytic number theory, connects to deep problems on irrationality/transcendence of constants, and links several classical arithmetic objects—Euler numbers, Bernoulli numbers, polygamma, and zeta function evaluations. Dirichlet beta values at integer arguments encode spectacular closed forms, integral representations, analytical continuations, and highly nontrivial behavior between even and odd indices.

1. Definition and Key Properties of the Dirichlet Beta Function

The Dirichlet beta function is the $L$-series attached to the primitive character modulo 4:

$$\beta(s) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^s}$$

for $\Re(s) > 0$ (Tanabe et al., 2023, Kawalec, 2020, Idowu, 2012).

Special values at integers appear in the study of periodic Fourier expansions, evaluations of $L$-values for real quadratic fields, and as arithmetic constants such as Catalan's constant ($\beta(2)$).

For all $s \in \mathbb{C}$, the classical functional equation is:

$$\beta(1-s) = (2\pi)^{-s} \sin\left( \frac{\pi s}{2} \right) \Gamma(s) \beta(s)$$

(Yakubovich, 2024, Bhattacharyya, 2019).

2. Explicit Closed Forms: Odd and Even Arguments

Odd Arguments $s = 2k+1$

The classical formula for odd-integer values expresses $\beta(2k+1)$ via Euler numbers $E_{2k}$:

$$\beta(2k+1) = (-1)^k \frac{E_{2k}}{2^{2k+2} (2k)!} \pi^{2k+1}$$

(Tanabe et al., 2023, Idowu, 2012, Zudilin, 2018, Kawalec, 2020, Hu et al., 2018, Yakubovich, 2024, Kim, 2014). For example:

• $\beta(1) = \frac{\pi}{4}$
• $\beta(3) = \frac{\pi^3}{32}$
• $\beta(5) = \frac{5\pi^5}{1536}$
• $\beta(7) = \frac{61\pi^7}{184320}$

Equivalent formulations employ generalized Bernoulli numbers $B_{2k+1,\chi_4}$:

$$\beta(2k+1) = (-1)^{k+1}(2k+1)\frac{2^{2k-1}}{2(2k)!} B_{2k+1,\chi_4} \pi^{2k+1}$$

(Tanabe et al., 2023).

Even Arguments $s = 2k$

No rational multiple of $\pi^{2k}$ expression is known for $\beta(2k)$ except for $k=1$ (Catalan's constant $G=\beta(2)$). Even-indexed values are given by polygamma/series/integrals, e.g.:

$$\beta(2k) = \frac{(-1)^{k-1}}{2(2k-1)!} \int_0^\infty \frac{x^{2k-1}}{\cosh x} dx$$

(Zudilin, 2018, Kim, 2014, Yakubovich, 2024, Kawalec, 2020), and polygamma differences:

$$\beta(2m) = \frac{(-1)^{m-1}}{2(2m-1)!}\left[ \psi^{(2m-1)}\left(\frac{1}{4}\right) - \psi^{(2m-1)}\left(\frac{3}{4}\right) \right]$$

(Kawalec, 2020, Idowu, 2012). General rapidly convergent series exist for all even $\beta(2k)$ (Connon, 2010).

3. Integral, Series, and Analytic Representations

Closed-form, recursive, and accelerated series have been established for various $\beta$ values. Central methodologies include:

• Integral Representations:
  • For odd $s=2k+1$: Integrals over Euler polynomials,

  $$I(k,m) := \int_0^{1/2} E_{2k}(t) \sin((2m+1)\pi t) dt,$$

  with the evaluation

  $I(k,m) = \frac{(-1)^k (2k)!}{(2m+1)^{2k+1}},$

  yielding

  $$\beta(2k+1) = (-1)^k \frac{E_{2k}}{2(2k)!} \pi^{2k+1}$$

  (Tanabe et al., 2023). - For even $s=2k$:

  $$\beta(2k) = -(-1)^{k-1} \frac{\pi^{2k}}{2^{2k-1}(2k-1)!} \int_0^{1/2} E_{2k-1}(t) \sec(\pi t) dt.$$

• Rapidly Convergent Series – Even Arguments:

$$\beta(2k) = \frac{(-1)^{k-1}}{2^{4k-1}(k-1)!} \sum_{n=1}^\infty \frac{1}{n(16 n^2 - 1)^k}$$

(Connon, 2010).

• Ramanujan-type Series:

For all $n\geq 1$,

$$\beta(2n) = \frac{1}{2(2n-1)!} \sum_{k=n}^{\infty} \frac{(2k)!}{2^{2k-1}(2k-2n)!} \zeta(2k)$$

$$\beta(2n+1) = \frac{1}{(2n)!} \sum_{k=n+1}^\infty \frac{(2k-1)!}{2^{2k} (2k-2n-1)!} \zeta(2k+1)$$

(Yakubovich, 2024).

• Polygamma Formulas:

$$\beta(n) = \frac{(-1)^{n-1}}{2^n 4^n} \left[ \psi^{(n-1)}\left(\frac{1}{4}\right) - \psi^{(n-1)}\left(\frac{3}{4}\right) \right]$$

(Idowu, 2012).

• Functional Equations:

$$\beta(s) = \frac{1}{2^s - 1} [\zeta(s) - 2^{-s} \zeta(s)]$$

(Idowu, 2012, Kawalec, 2020).

4. Connections with Euler Numbers, Bernoulli Numbers, and the Zeta Function

Dirichlet beta values at odd integers are directly tied to Euler numbers $E_{2n}$; all even-indexed Euler numbers and polynomials appear in generating functions and in explicit formulas. For Bernoulli numbers, the connection arises via zeta values and through identities for the generalized Bernoulli numbers $B_{2k+1,\chi_4}$:

$E_{2k} = - (2k+1) 2^{2k-1} B_{2k+1,\chi_4}$

(Tanabe et al., 2023).

Polygamma differences generate all $\beta(n)$ via repeated derivatives of $\cot(\pi z)$ at $z=1/4$:

$$\beta(n) = \frac{\pi}{2^n n!} \left. \frac{d^{n-1}}{dz^{n-1}} \cot(\pi z) \right|_{z=1/4}$$

(Idowu, 2012).

Table: Comparison of Key Dirichlet Beta Value Representations

Argument	Closed Form	Integral/Series Representation
$\beta(2k+1)$	$(-1)^k\dfrac{E_{2k}}{2^{2k+2} (2k)!}\pi^{2k+1}$	$\int_0^\infty \dfrac{t^{2k}}{\cosh t} dt$ (Zudilin, 2018)
$\beta(2k)$	None in $\pi^{2k}$ terms; e.g. $G$ for $k=1$	$$\dfrac{(-1)^{k-1}}{2(2k-1)!}\int_0^\infty \dfrac{x^{2k-1}}{\cosh x} dx$$
General $n$	Polygamma differences	$$\dfrac{(-1)^{n-1}}{2^n 4^n}[\psi^{(n-1)}(1/4)-\psi^{(n-1)}(3/4)]$$

5. Arithmetic, Irrationality, and Transcendence Phenomena

The arithmetic of Dirichlet beta values is a subject of open conjectures and partial results:

• Infinite irrationality: At least one of $\beta(2), \beta(4),\dots,\beta(12)$ is irrational (Zudilin, 2018); indeed, infinitely many $\beta(2n)$ are irrational (Zudilin, 2018).
• No closed form in pure powers of $\pi$ is known for $\beta(2n)$ for $n\ge2$, and it is widely conjectured that none exists.
• Even values such as $\beta(2)$ (Catalan's constant, $G$) remain not known to be transcendental.
• Special sequences and convolution identities in $L$-series (e.g., Williams's convolution) relate $\beta(2k+1)$ and $\beta(2k)$ (Hu et al., 2018).

6. Generalizations and Advanced Methodologies

Recent research introduces parametric generalizations such as the two-variable Dirichlet beta function,

$\beta(z,s) = \frac{1}{\Gamma(s)} \int_0^\infty x^{s-1} \sech^z(x) dx$

(Yakubovich, 2024), with functional equations, recurrence relations, and master Ramanujan-type identities unifying entire families of $L$- and zeta values.

Alternative approaches include creative telescoping (WZ-methods) (Chen, 2012), accelerated series via theta and Dedekind $\eta$-function transforms (Patkowski, 2015), and analytic continuation/regularization of divergent sums at negative arguments via anti-limit extrapolation, leading to

$\beta(-n) = \frac{E_n}{2}$

for $n=0,1,2,\dots$ (Bhattacharyya, 2019, Hu et al., 2018).

7. Worked Out Examples and Numerical Values

Notable explicit cases (from several sources):

$s$	Value	Numeric Approximation
$1$	$\frac{\pi}{4}$	$0.785398...$
$2$	Catalan’s $G$	$0.915965...$
$3$	$\frac{\pi^3}{32}$	$0.968946...$
$4$	No closed form; polygamma/integral	$0.988945...$
$5$	$\frac{5\pi^5}{1536}$	$0.996157...$
$6$	(as above)	$0.999053...$

For negative arguments,

$\beta(-2m) = \frac{E_{2m}}{2}$

e.g., $\beta(-2) = -\frac{1}{2}$, $\beta(-4) = \frac{5}{2}$ (Bhattacharyya, 2019).

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Dirichlet beta values occupy a central place in the landscape of special $L$-values and transcendental number theory, with deep connections to Euler numbers, Bernoulli numbers, analytic continuation, and modern generalizations. Their explicit formulas, arithmetical structure, and rapidly converging representations have inspired a broad spectrum of contemporary research methodologies, with ongoing investigations into their irrationality and transcendence.