Closed-Form Evaluation of Arctanh Power Sums via Infinite Products

This paper establishes closed-form expressions for the infinite series of arctanh(n⁻ᵏ) by linking them to gamma function infinite products, subsequently deriving new identities for the Riemann zeta function and Euler-Mascheroni constant, and analyzing the monotonicity, convexity, and asymptotic behavior of these sums.

The Gist

Imagine you are trying to measure the "distance" between a number and its neighbors in an infinite line of numbers. In mathematics, we often add up tiny pieces to find a total sum. This paper is about a very specific, tricky kind of sum involving a function called arctanh (which is related to how fast things grow or shrink).

The author, Ryan Goulden, asks a simple question: If we add up these tiny pieces forever, do we get a neat, clean answer, or just a messy, endless decimal?

Here is the story of the paper, broken down into everyday concepts:

1. The Infinite Puzzle

Imagine you have a giant jar of marbles. Each marble has a number on it ($n = 2, 3, 4, \dots$). You want to calculate a specific value for each marble based on a rule involving a power $k$ (like squaring it, cubing it, etc.).

The paper looks at the sum of these values as you keep adding more and more marbles forever.

• The Problem: Usually, these infinite sums are messy. They don't have a simple "closed-form" answer (like $\pi$ or $\sqrt{2}$). They just look like $0.650923\dots$ and go on forever.
• The Discovery: Goulden found a "magic key" (a formula involving infinite products) that unlocks these sums. He showed that for every integer $k \ge 2$, this messy sum actually equals a very clean, exact number.

2. The "Doubling" Analogy

To solve the puzzle, the author uses a clever trick involving infinite products (multiplying numbers forever instead of adding them).

Think of it like a recipe.

• You have a base ingredient called $g(k)$.
• The author discovered a "doubling law": If you take the ingredient for $k$ and double the power to $2k$, the new ingredient isn't just twice as big. It's related to the square of the original ingredient, but with a little "extra" factor.
• That "extra factor" is exactly the sum we are trying to find!
• The Metaphor: Imagine you have a cake. If you double the size of the cake, it doesn't just get twice as big; it gets a specific amount of extra frosting. The paper calculates exactly how much "frosting" (the sum) is needed to make the math work.

3. The Special Case: The "Golden" Number ($k=3$)

The paper finds that most of these sums result in numbers that are "transcendental" (numbers like $\pi$ or $e$ that can't be written as simple fractions).

However, there is one unique case: $k=3$.

• For $k=3$, the messy sum simplifies to a beautiful, rational fraction: $\frac{1}{2} \ln(1.5)$.
• The Analogy: Imagine you are walking through a forest of strange, alien trees. Most trees are made of glass or smoke. But one specific tree (the $k=3$ tree) is made of solid, familiar oak. It's the only one that follows a simple, rational rule.
• Because this number is so special, the author proves it is "transcendental" (a fancy way of saying it's a unique, non-repeating number that can't be built from simple fractions).

4. Fixing the "Broken" Constant ($\gamma$)

The paper also tackles the Euler-Mascheroni constant ($\gamma$), a famous number in math that appears in many places (like prime numbers and logarithms).

• The Old Way: Calculating $\gamma$ is like trying to fill a bucket with a leaky hose. You have to pour in millions of drops (terms) to get just a few digits of accuracy. It's slow and frustrating.
• The New Way: Goulden found a "super-hose." By rearranging the math using the same "infinite product" tricks, he created a new formula that fills the bucket exponentially faster.
• The Result: With the old method, you might need 100,000 drops to get 5 correct digits. With the new method, you only need 20 drops to get 13 correct digits. It's like switching from a drip-feed to a firehose.

5. Why Does This Matter?

• Structure: It connects two different worlds of math: the world of sums (adding numbers) and the world of products (multiplying numbers). It shows they are two sides of the same coin.
• Speed: It gives mathematicians a much faster way to calculate important constants like $\gamma$ and the Riemann Zeta function (which is famous for the unsolved "Riemann Hypothesis").
• Beauty: It proves that even in the infinite chaos of numbers, there are hidden patterns and simple answers waiting to be found, especially for the number 3.

Summary

Ryan Goulden took a complicated, infinite math problem and showed that it has a hidden, elegant structure. He found a "magic formula" that turns a messy sum into a clean answer, discovered that the number 3 is the "special child" of this family, and invented a super-fast way to calculate one of math's most famous constants. It's a story of finding order in infinity.

Technical Summary

1. Problem Statement

The paper addresses the evaluation of the infinite series:
$$h(k) := \sum_{n=2}^\infty \text{arctanh}(n^{-k})$$
for integers $k \ge 2$. While series involving the Riemann zeta function ($\sum n^{-s}$) are well-understood, series involving inverse hyperbolic functions like $\text{arctanh}$ have received less systematic treatment. The primary goal is to establish explicit closed-form expressions for $h(k)$ and explore their arithmetic and analytic properties.

2. Methodology

The author employs a multi-faceted approach combining classical analysis, infinite product theory, and transcendental number theory:

• Logarithmic Transformation: The core technique involves converting the $\text{arctanh}$ terms into logarithmic ratios using the identity $\text{arctanh}(x) = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$. This transforms the sum into the logarithm of an infinite product.
• Infinite Product Identities: The resulting products are identified with known infinite products $f(k)$ and $g(k)$ defined as:
  $$f(k) = \prod_{n=1}^\infty \left(1 + \frac{1}{n^k}\right), \quad g(k) = \prod_{n=2}^\infty \left(1 - \frac{1}{n^k}\right)$$
  The paper relies on Cantrell's formulas (derived from Weierstrass factorization, Euler reflection, and Gauss multiplication formulas for the Gamma function) to evaluate these products in terms of $\Gamma(z)$ (for odd $k$) or trigonometric functions (for even $k$).
• Series Manipulation: The Fubini–Tonelli theorem is used to interchange summation orders, allowing the derivation of zeta-series representations from the Taylor expansion of $\text{arctanh}$.
• Integral Theorems: Frullani's integral theorem is utilized to derive a new series representation for the Euler–Mascheroni constant ($\gamma$).
• Transcendence Theory: Baker's theorem and Nesterenko's theorem are applied to determine the arithmetic nature (rationality, irrationality, transcendence) of specific values.

3. Key Contributions and Results

A. Closed-Form Evaluation (Theorem 1)

The paper establishes the main result:
$$h(k) = \frac{1}{2} \ln\left( \frac{g(2k)}{g(k)^2} \right) = \frac{1}{2} \ln\left( \frac{f(k)}{2g(k)} \right)$$
This connects the sum directly to the deviation of the product $g(k)$ from a "doubling law" ($g(2k) = g(k)^2 e^{2h(k)}$). Explicit formulas are provided for $f(k)$ and $g(k)$ based on the parity of $k$:

• Odd $k$: Expressed via finite products of Gamma functions.
• Even $k$: Expressed via finite products of trigonometric functions (sines).

B. Zeta-Series Representation (Proposition 9)

The sum is shown to admit an equivalent series representation involving the Riemann zeta function:
$$h(k) = \sum_{m=0}^\infty \frac{\zeta((2m+1)k) - 1}{2m+1}$$
This series converges absolutely and provides a rapid way to compute $h(k)$ for large $k$.

C. Structural Identity for $\zeta(k)$ (Corollary 2)

A novel identity relating the Riemann zeta function to $h(k)$ is derived:
$$\zeta(k) = 1 + h(k) - C(k)$$
where $C(k)$ is a rapidly convergent correction term ($O(2^{-3k})$). This decomposes $\zeta(k)$ into a dominant arctanh-sum and a negligible correction.

D. New Representation for Euler–Mascheroni Constant (Corollary 3 & Theorem 25)

The paper derives a new series for $\gamma$:
$$\gamma = 1 - \frac{1}{2}\ln 2 - \sum_{n=2}^\infty \left( \text{arctanh}(n^{-1}) - n^{-1} \right)$$
Crucially, the author accelerates this series using the zeta-representation to achieve geometric (exponential) convergence:
$$\gamma = 1 - \frac{1}{2}\ln 2 - \sum_{m=1}^\infty \frac{\zeta(2m+1) - 1}{2m+1}$$
This accelerated form is significantly more efficient for numerical computation than the original polynomially convergent series.

E. Arithmetic Properties and Transcendence (Theorem 29 & 32)

• Uniqueness of $k=3$: The paper proves that $k=3$ is the unique integer $k \ge 2$ for which the ratio $f(k)/(2g(k))$ is rational.
  • Specifically, $h(3) = \frac{1}{2}\ln(3/2)$.
  • By Baker's theorem, since $3/2$ is rational and not 1, $h(3)$ is transcendental.
• Apéry's Constant: This yields a representation for $\zeta(3)$:
  $$\zeta(3) = 1 + \frac{1}{2}\ln(3/2) - C(3)$$
  where $C(3)$ converges geometrically.
• Other Values: For even $k$, the values involve $\sinh(\pi)$ and are shown to be transcendental (via Nesterenko's theorem). For odd $k \ge 5$, the arithmetic nature remains an open problem, though the product structure suggests they are likely transcendental.

F. Analytic Properties (Section 7)

The function $h(k)$ is proven to be strictly decreasing and strictly convex for $k > 1$. The paper provides:

• Explicit asymptotic expansions as $k \to \infty$.
• Tight two-sided bounds for $h(k)$ using partial sums of the zeta series.
• Analysis of the forward differences $\Delta h(k)$.

4. Significance

• Unification: The work unifies inverse hyperbolic sums, infinite products, and the Riemann zeta function, revealing deep structural connections between them.
• Computational Efficiency: The derivation of the exponentially convergent series for $\gamma$ offers a practical improvement over existing methods, allowing high-precision calculation with far fewer terms.
• Number Theory: The identification of $k=3$ as the unique case yielding a rational product ratio is a significant number-theoretic result, linking cyclotomic polynomials ($\Phi_3$ and $\Phi_6$) to the telescoping nature of the infinite product.
• New Constants: The paper catalogs decimal expansions of $h(k)$ and related sequences, suggesting new entries for the OEIS (e.g., A393668 for $2h(2)$).

In summary, Goulden provides a comprehensive analytical framework for evaluating a specific class of infinite series, yielding closed forms, new identities for fundamental constants, and rigorous proofs of transcendence for specific cases.