Arctanh Sums: Analytic Continuation and Prime-Restricted Theory

This paper investigates the analytic continuation and prime-restricted theory of arctanh sums $h(k)$, establishing their meromorphic extension with specific poles and zeros, deriving Laurent expansions and Mittag-Leffler decompositions, and proving the unconditional transcendence of their prime-restricted analogues $h_p(2j)$ via a $\pi$-cancellation mechanism.

The Gist

Imagine you have a giant, infinite machine that takes a number, $k$, and spits out a result by adding up an endless list of tiny calculations. This machine is called $h(k)$.

In the world of advanced math, this machine is usually only allowed to work when $k$ is a "big" number (specifically, greater than 1). If you try to feed it a small number, the machine explodes because the sum goes to infinity.

This paper by Ryan Goulden is like a master mechanic who has figured out how to fix the machine so it can run on small numbers, and then discovered a secret "prime-only" version of the machine that reveals deep secrets about the universe of numbers.

Here is the breakdown of what the paper does, using simple analogies:

1. The "Broken" Machine and the Magic Fix (Analytic Continuation)

The Problem: The original machine, $h(k)$, is defined by adding up terms like $\text{arctanh}(1/n^k)$. For big $k$, the terms get tiny fast, and the sum works. But for small $k$ (like $k=0.5$), the terms are too big, and the sum explodes.

The Fix: The author uses a mathematical "time machine" (called Analytic Continuation). He finds a secret formula that looks different but gives the same result for big numbers. Because this new formula is smoother, it doesn't explode when $k$ gets small.

• The Result: The machine now works for almost any positive number. However, it has "potholes" (called poles) at specific spots like $k=1, 1/3, 1/5$, etc. If you drive the car right over these spots, the engine blows up. But between the potholes, the road is smooth.

2. The "Ghost" Zeros (Where the Machine Hits Zero)

Between every two potholes on the road, the machine's output goes from "infinity" to "negative infinity."

• The Discovery: Because it goes from positive to negative, it must cross zero somewhere in between. The author proves that in every single gap between the potholes, the machine crosses zero exactly once.
• The Analogy: Imagine a rollercoaster that goes up a cliff, drops down, and goes up another cliff. It crosses the ground level exactly once between the hills. These "zero crossings" are special points that tell us something about the famous Riemann Zeta function (a giant mathematical puzzle about prime numbers).

3. The "Prime-Only" Machine (The Multiplicative Theory)

The original machine sums over every number (2, 3, 4, 5, 6...).
The author builds a new, smaller machine called $h_p(k)$ that only sums over Prime Numbers (2, 3, 5, 7, 11...).

• Why it matters: Primes are the building blocks of all numbers. By isolating them, the author creates a "pure" version of the math.
• The Magic Trick: When you plug in even numbers (like 2, 4, 6) into this prime-only machine, something magical happens. The messy, irrational parts of the answer (involving $\pi$) cancel each other out perfectly, leaving behind a clean, rational number inside a logarithm.
• The Big Win: Because the answer is a clean rational number inside a log, the author can prove (using a famous theorem by Baker) that these answers are Transcendental.
  • Translation: These numbers are so weird and complex that they cannot be the solution to any simple algebraic equation. They are "mathematically unique" in a very strong sense.

4. The "Zero Sum" Connection (The Hadamard Product)

The most exciting part of the paper connects the prime-only machine to the non-trivial zeros of the Riemann Zeta function.

• The Analogy: Think of the Riemann Zeta function as a giant orchestra. The "zeros" are the specific notes where the music stops (silence).
• The Discovery: The author shows that the value of the prime-only machine ($h_p$) can be calculated by adding up a specific contribution from every single one of those silent notes in the orchestra.
• The Benefit: Usually, adding up these notes is messy and slow. But because of the way the author built the machine (subtracting a specific part), the messy parts cancel out. The sum converges very quickly, making it much easier to calculate and study these mysterious zeros.

5. The "Odd/Even" Split

The paper highlights a funny split in the math world:

• Even Numbers: When you use even numbers in the prime-only machine, the math is clean, the $\pi$'s cancel out, and we can prove the results are transcendental.
• Odd Numbers: When you use odd numbers, the $\pi$'s don't cancel. The math gets messy, and we don't know if the results are transcendental yet.
• Why it's cool: This mirrors a famous mystery in math: we know exactly what $\zeta(2), \zeta(4), \zeta(6)$ are (they involve $\pi$), but we still don't fully understand $\zeta(3), \zeta(5)$, etc. This paper shows that this "Even vs. Odd" mystery is built right into the structure of these sums.

Summary

Ryan Goulden took a broken math machine, fixed it so it works on small numbers, found exactly where it crosses zero, and then built a "Prime-Only" version that acts like a super-sensor. This sensor can detect the hidden "silent notes" (zeros) of the Riemann Zeta function and prove that certain combinations of prime numbers result in numbers so complex they are mathematically unique.

It's like taking a blurry photo of a complex pattern, sharpening the focus, and realizing that the pattern is actually a perfect map of the most famous unsolved puzzle in mathematics.

Technical Summary

Here is a detailed technical summary of the paper "ARCTANH SUMS: ANALYTIC CONTINUATION AND PRIME-RESTRICTED THEORY" by Ryan Goulden.

1. Problem Statement

The paper investigates the analytic properties and arithmetic structure of the infinite series defined by the sum of inverse hyperbolic tangents:
$$h(k) = \sum_{n=2}^{\infty} \text{arctanh}(n^{-k})$$
Initially defined for $\text{Re}(k) > 1$, the author seeks to:

1. Establish the meromorphic continuation of $h(k)$ to the right half-plane ($\text{Re}(k) > 0$).
2. Analyze the zero structure (location and uniqueness of real zeros) of the continued function.
3. Develop a prime-restricted analogue, $h_p(k)$, summing only over prime indices, and explore its connection to the Riemann zeta function $\zeta(s)$, specifically regarding transcendence and product formulas involving non-trivial zeros.

This work serves as Part II and III of a three-part program, building upon a companion preprint (arXiv:2602.06244) that established closed-form evaluations for integer $k \ge 2$.

2. Methodology

The author employs a combination of analytic number theory and complex analysis techniques:

• Zeta-Series Representation: Utilizing the identity $h(k) = \sum_{m=0}^{\infty} \frac{\zeta((2m+1)k) - 1}{2m+1}$, derived from the Taylor expansion of $\text{arctanh}$.
• Mittag-Leffler Decomposition: Separating the function into a polar part (sum of principal parts at poles) and a holomorphic remainder to study singularities and asymptotic behavior.
• Hadamard Product Factorization: Applying the canonical product representation of the completed zeta function $\xi(s)$ to derive product formulas for the prime-restricted sum.
• Möbius Inversion: Utilizing the divisibility properties of odd integers to invert the relationship between $h(k)$ and $\zeta(k)$.
• Transcendence Theory: Applying Baker's Theorem on linear forms in logarithms of algebraic numbers to prove the transcendence of specific values.

3. Key Contributions and Results

Part I: Analytic Continuation and Pole Structure

• Meromorphic Extension: The function $h(k)$ extends meromorphically to $\text{Re}(k) > 0$.
• Singularities: The function possesses simple poles at $k_m = \frac{1}{2m+1}$ for $m = 0, 1, 2, \dots$.
  • The residue at each pole is $\text{Res}_{k=k_m} h(k) = \frac{1}{(2m+1)^2}$.
  • The sum of all residues is $\sum (2m+1)^{-2} = \frac{\pi^2}{8}$.
  • The poles accumulate at $k=0$, making it a non-isolated singularity; thus, no analytic continuation exists to $\text{Re}(k) < 0$.
• Laurent Expansions:
  • At $k=1$, the expansion is $h(k) = \frac{1}{k-1} - \frac{1}{2}\ln 2 + O(k-1)$. The finite part (regularized value) is $-\frac{1}{2}\ln 2$.
  • General expansions at $k_m$ involve Euler-Mascheroni constants and sums of $\zeta$ values at rational arguments.
• Mittag-Leffler Decomposition: $h(k)$ is decomposed into a polar part $h_{\text{polar}}(k)$ (encoding the Dirichlet lambda function $\lambda(s)$) and a holomorphic remainder $\phi(k)$.

Part II: Zero Structure and Duality

• Uniqueness of Zeros: On each interval between consecutive poles, $I_n = (\frac{1}{2n+3}, \frac{1}{2n+1})$, the function $h(k)$ is strictly decreasing and has exactly one simple real zero $z_n$.
• Asymptotics: The zeros satisfy $z_n = \frac{1}{2n+2} + O(n^{-3})$. As $n \to \infty$, $\zeta(z_n) \to -1/2$.
• Pole-Zero Duality: The zeros of $h(k)$ correspond to points where $\zeta(k) = 1 - C(k)$, establishing a duality between the zeros of $h$ and the values of $\zeta$.

Part III: Prime-Restricted Theory ($h_p$)

The author defines the prime-restricted sum:
$$h_p(k) = \sum_{p \text{ prime}} \text{arctanh}(p^{-k})$$

• Closed Form: $h_p(k) = \ln \zeta(k) - \frac{1}{2} \ln \zeta(2k) = \frac{1}{2} \ln \left( \frac{\zeta(k)^2}{\zeta(2k)} \right)$.
• Transcendence at Even Integers: For any integer $j \ge 1$, $h_p(2j)$ is unconditionally transcendental.
  • Mechanism: The $\pi$ factors in $\zeta(2j)$ and $\zeta(4j)$ cancel out exactly in the combination $\zeta(2j)^2 / \zeta(4j)$, leaving a rational number $r_j$. Thus, $h_p(2j) = \frac{1}{2} \ln r_j$. By Baker's theorem, the logarithm of a rational $r_j > 1$ is transcendental.
• Product Formula over Zeros: Using the Hadamard product for $\xi(s)$, the author derives a product formula for $h_p(k)$ involving the non-trivial zeros $\rho$ of $\zeta(s)$:
  $$h_p(k) = \sum_{\rho} \left[ \ln\left(1 - \frac{k}{\rho}\right) - \frac{1}{2} \ln\left(1 - \frac{2k}{\rho}\right) \right] + E(k)$$
  • Significance: The linear regularization terms ($k/\rho$) cancel out in this specific combination, resulting in a sum that converges absolutely with $O(|\text{Im}(\rho)|^{-2})$ decay. This improves upon standard formulas for $-\zeta'/\zeta$ which require symmetric pairing or conditional convergence.
• Möbius Inversion: The paper establishes an additive inversion formula recovering $\zeta(k)-1$ from $h(k)$ by summing over odd integers:
  $$\zeta(k) - 1 = \sum_{d \text{ odd}} \frac{\mu(d)}{d} h(dk)$$
  This provides a method to compute $\zeta(3)$ (Apéry's constant) using values of $h$ at higher odd integers.

4. Significance and Implications

• Analytic Insight: The paper provides a complete description of the singularities and zero distribution of a non-standard Dirichlet-type series, revealing a dense accumulation of poles at the origin.
• Transcendence: It offers a new, unconditional proof of the transcendence of specific combinations of zeta values at even integers via the "pi-cancellation" mechanism in the prime-restricted sum.
• Zeta Function Connection: The work deepens the link between additive sums (over integers) and multiplicative structures (over primes). The prime-restricted function $h_p$ acts as a "filter" that isolates the odd-power components of $\ln \zeta$, making it sensitive to the non-trivial zeros of $\zeta$ in a way that the full sum $h$ is not.
• Computational Utility: The derived product formula with $O(|\gamma|^{-2})$ convergence offers a potentially more efficient way to approximate values related to the Riemann zeros compared to standard logarithmic derivative formulas.
• Open Questions: The paper concludes with open problems regarding the holomorphic extension of the remainder term $\phi(k)$ past $\text{Re}(k)=0$ and the transcendence of finite parts $A_m$ involving $\zeta$ at rational arguments.

In summary, Goulden's work bridges complex analysis and arithmetic by rigorously analyzing the analytic continuation of arctanh sums, uncovering a rich structure of poles and zeros, and leveraging prime restrictions to derive new transcendence results and convergent product formulas for the Riemann zeta function.