Stringent bounds for the non-zero Bernoulli numbers

Technical Summary: Stringent Bounds for the Non-zero Bernoulli Numbers

Problem Statement
The paper addresses the problem of establishing sharp lower and upper bounds for the absolute values of non-zero even-indexed Bernoulli numbers, denoted as $|B_{2k}|$. While various bounds exist in the literature, including those derived from the Riemann zeta function and Stirling's formula, the author identifies a need for refined inequalities. Specifically, the paper notes that while upper bounds have been sharpened (e.g., by H. Alzer), the lower bounds in existing double inequalities remain less precise than they could be. The objective is to determine the best possible constants $\alpha$ and $\beta$ to define a double inequality that strictly bounds $|B_{2k}|$ for all $k \in \mathbb{N}$.

Methodology
The author employs Euler's product formula for the Riemann zeta function, $\zeta(s)$, as the primary analytical tool. The methodology proceeds through the following logical steps:

1. Foundational Relations: The paper utilizes the known relation between Bernoulli numbers and the zeta function:
   $$|B_{2k}| = \frac{2 \cdot (2k)!}{(2\pi)^{2k}} \zeta(2k)$$
   and the Euler product representation:
   $$\zeta(2k) = \left(\frac{2^{2k}}{2^{2k}-1}\right) \prod_{n=1}^{\infty} \frac{p_n^{2k}}{p_n^{2k}-1}$$
   where $p_n$ represents the sequence of odd prime integers ($3, 5, 7, \dots$).

2. Derivation of Lower Bounds: By truncating the infinite product in the Euler formula after $m$ terms, the author establishes a family of lower bounds. It is demonstrated that the product form involving primes provides a sharper lower bound than previous summation-based bounds (such as those derived from Fourier representations).

3. Monotonicity Analysis: To find the optimal constants for the specific double inequality involving the term $3^{2k}$, the author defines a function $h(x)$:
   $$h(x) = 3^x \left[ 1 - \frac{1}{\left(1 - \frac{1}{2^x}\right)\zeta(x)} \right]$$
   The proof of the main theorem relies on establishing that $h(x)$ is strictly decreasing for $x \in \{2, 4, 6, \dots\}$. This is achieved by analyzing the difference $g(x) = h(x) - h(x+1)$ and expanding the terms using series representations of the zeta function to show $g(x) > 0$.

4. Determination of Constants: The limits of $h(x)$ as $x \to \infty$ and the value at $x=2$ are calculated to determine the tightest possible constants $\alpha$ and $\beta$.

Key Contributions and Results
The paper presents the following main results:

• Refined Lower Bounds: The author proves that for any $m, k \in \mathbb{N}$ and primes $p_n$, the inequality
  $$\frac{2 \cdot (2k)!}{\pi^{2k}(2^{2k}-1)} \prod_{n=1}^{m} \frac{p_n^{2k}}{p_n^{2k}-1} < |B_{2k}|$$
  holds. This generalizes previous results and is shown to be sharper than bounds derived from partial sums of odd reciprocals.

• Optimal Double Inequality: The central result is Theorem 2.6, which establishes the best possible constants $\alpha$ and $\beta$ for the double inequality:
  $$\frac{2 \cdot (2k)!}{\pi^{2k}(2^{2k}-1)} \frac{3^{2k}}{(3^{2k}-\alpha)} < |B_{2k}| < \frac{2 \cdot (2k)!}{\pi^{2k}(2^{2k}-1)} \frac{3^{2k}}{(3^{2k}-\beta)}$$
  The constants are determined to be:

  • $\alpha = 1$
  • $\beta = 9\left(1 - \frac{8}{\pi^2}\right) \approx 1.704875$

• Comparison with Existing Bounds: The paper demonstrates that the new bounds are strictly stronger than those provided by H. Alzer (specifically inequality 1.9 in the text) and H.-F. Ge. Numerical analysis (Table 1) and graphical plots (Figure 1) illustrate that the difference between the new bounds and the actual values of $|B_{2k}|$ is significantly smaller than in previous formulations.

Significance and Claims
The author claims that the double inequality (2.7) improves upon all existing bounds for non-zero Bernoulli numbers found in the literature. The significance of this work lies in:

1. Precision: Providing the "best possible" constants for a specific class of inequalities involving the term $3^{2k}$.
2. Application Potential: Given the role of Bernoulli numbers in the series expansions of trigonometric and hyperbolic functions, the author posits that these refined inequalities will be useful in deriving "fine inequalities" involving these functions.
3. Future Direction: The paper concludes with a conjecture extending the main result to a general product form involving $m$ primes, suggesting that the method used to prove the monotonicity of $h(x)$ could be applied to determine optimal constants for broader classes of inequalities.

The paper maintains a modest tone, focusing strictly on the mathematical derivation and refinement of inequalities without proposing new experimental applications or broader theoretical frameworks beyond the scope of bounding $|B_{2k}|$.