Elementary proof of some Ramanujan-type identities

This paper provides an elementary proof for Ramanujan-type identities expressing the squares of the Riemann zeta function at integer points through series involving hyperbolic functions, the digamma function, and Bernoulli numbers, while also correcting previous inaccuracies and adding new sections, a figure, and a reference.

The Gist

Imagine you are trying to solve a massive, cosmic puzzle. The pieces of this puzzle are numbers, specifically the Riemann Zeta function (let's call it $\zeta$). This function is famous for being the "heartbeat" of prime numbers, but it's also incredibly complex.

The paper you're looking at is like a master craftsman showing you a new, simple tool to rearrange the pieces of this puzzle. The author, M.A. Korolev, isn't using a sledgehammer (heavy, complex math); he's using a leverage tool (a simple, elementary identity) to prove some very deep, surprising connections between different parts of the number world.

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

1. The Magic Lever (The Main Idea)

The heart of the paper is a simple trick called Lemma 1.

Imagine you have a pile of stones (numbers). You want to know the total weight of every possible pair of stones you can pick.

• The Hard Way: Pick every stone, pair it with every other stone, weigh them, and add them up.
• The Magic Way: The author shows that if you arrange the stones in a specific way, the total weight of all pairs is exactly half the square of the total weight of the whole pile.

It sounds like magic, but it's just a clever rearrangement of terms. The author uses this "magic lever" to take a messy, infinite sum of numbers and turn it into a clean, elegant formula.

2. The Goal: Squaring the Circle (Literally, Sort of)

The paper's main goal is to find the square of the Zeta function ($\zeta^2$).

• Think of $\zeta(2)$ as a specific number (about 1.645).
• $\zeta^2(2)$ is that number multiplied by itself.
• Usually, calculating the square of these infinite series is a nightmare.

Korolev uses his "Magic Lever" to show that these squares aren't just random numbers. They are actually equal to new, faster-converging series involving:

• Hyperbolic functions: Think of these as "curved" versions of the familiar sine and cosine waves, often used to describe hanging chains or heat flow.
• The Digamma function: A "smoothed-out" version of the harmonic series (1 + 1/2 + 1/3...).
• Bernoulli numbers: A special family of numbers that pop up everywhere in calculus and number theory, like secret agents.

3. The Analogy of the "Infinite Mirror"

One of the most beautiful parts of the paper is how it connects two different worlds:

1. The World of Sums: Adding up infinite lists of fractions (like $1/1^2 + 1/2^2 + \dots$).
2. The World of Functions: Using complex curves and waves to describe those same sums.

The author shows that if you look at the "square" of the sum in one world, you can see it reflected perfectly in the other world using these hyperbolic functions. It's like looking at a mountain in a lake; the reflection (the new series) is just as real as the mountain, but it gives you a different perspective that is easier to measure.

4. Why is this "Elementary"?

In math, "elementary" doesn't mean "easy for a child." It means it doesn't require the most advanced, heavy machinery (like complex contour integration or heavy machinery from modern analysis).

The author is saying: "You don't need a rocket ship to get to the moon. You just need a really good ladder."
He uses basic algebra and a few standard rules about infinite series to prove things that usually require PhD-level tools. This makes the proof more accessible and reveals the underlying "simple logic" hidden inside the complex formulas.

5. The "Ramanujan" Connection

The title mentions Ramanujan. Srinivasa Ramanujan was a genius who wrote down hundreds of formulas that looked like magic. He often knew the answer but couldn't always explain the "why."

This paper is like a detective story where the author finds the "why." He takes some of Ramanujan's mysterious formulas and says, "Here is the simple, logical path that leads to these results." He proves that these strange identities aren't just accidents; they are the natural result of applying that simple "Magic Lever" (Lemma 1) to the right problems.

Summary: What did they actually find?

The paper provides a recipe book. If you want to calculate the square of a Zeta number (like $\zeta(3)^2$ or $\zeta(5)^2$), you can now use these new formulas.

• Old way: Summing a very slow, messy series.
• New way: Using these new formulas involving hyperbolic functions and digamma functions, which converge (settle down to an answer) much faster.

In a nutshell: The author took a simple algebraic trick, applied it to the most famous number series in the world, and discovered a new, faster, and more beautiful way to calculate their squares, connecting them to the geometry of waves and the secrets of prime numbers. It's a reminder that even in the most complex parts of mathematics, there is often a simple, elegant truth waiting to be found.

Technical Summary

Here is a detailed technical summary of the paper "Elementary proof of some Ramanujan-type identities" by M.A. Korolev.

1. Problem Statement

The paper addresses the problem of establishing new identities that relate the squares of the Riemann zeta function at integer points, $\zeta^2(s)$, to rapidly convergent series involving:

• Hyperbolic functions (e.g., $\coth$, $\sinh$).
• The Digamma function ($\psi(z)$).
• Bernoulli numbers and polynomials.
• Arithmetic functions (divisor functions, Möbius function, etc.).

While Ramanujan and others (Cauchy, Lerch) previously discovered specific instances of such identities (e.g., relating $\zeta(3)$ to series involving $e^{2\pi n}$), this work aims to provide a unified, elementary framework to derive these relations and generalize them to arbitrary integer arguments and Dirichlet series.

2. Methodology

The author employs a purely elementary analytic approach, avoiding complex contour integration in favor of algebraic manipulation and series rearrangement. The core methodology consists of the following steps:

A. The Fundamental Lemma

The foundation of the paper is Lemma 1, an identity for arbitrary sequences $\{a_n\}$ and $\{b_n\}$ (with $b_n > 0$):
$$\sum_{m,n=1}^N \frac{a_m a_n}{b_m(b_m + b_n)} = \frac{1}{2} \left( \sum_{n=1}^N a_n \right)^2$$
This identity allows the transformation of a double sum involving a denominator $(b_m + b_n)$ into the square of a single sum. By setting $a_n = n^{-s}$ and $b_n = n^k$, the author connects $\zeta^2(s)$ to double series of the form $\sum \frac{1}{n^k(m^k + n^k)}$.

B. Partial Fraction Decomposition

To evaluate the inner sums of the double series, the author utilizes Lemma 3 and Lemma 4, which provide partial fraction expansions for rational functions of the form $\frac{w^s}{w^{2k} + 1}$ and $\frac{w^s}{w^{2k+1} + 1}$. These expansions involve roots of unity ($\varepsilon_r$ and $\omega_r$) and allow the decomposition of terms like $\frac{1}{m^k + n^k}$ into linear terms involving $m \pm n\varepsilon_r$.

C. Summation and Special Functions

Using the partial fraction decomposition, the double series are reduced to single series involving the Digamma function $\psi(z)$. The author utilizes standard properties of $\psi(z)$, including:

• Series representation: $\psi(z) = -\gamma - \frac{1}{z} + \sum (\frac{1}{n} - \frac{1}{n+z})$.
• Reflection formula: $\psi(-z) = \psi(z) + \frac{1}{z} + \pi \cot(\pi z)$.
• Asymptotic expansions involving Bernoulli numbers (Lemma 9).

D. Generalization via Dirichlet Convolution

In Section 4, the method is generalized to arbitrary Dirichlet series. By introducing an arithmetic function $g(m)$ and its Dirichlet convolution $f(n) = \sum_{d|n} g(d)$, the author derives identities for $L^2(s; f)$, where $L(s; f)$ is the Dirichlet series associated with $f$.

3. Key Contributions and Results

A. New Identities for $\zeta^2(s)$

The paper derives three main theorems expressing $\zeta^2(s)$ as series involving $\psi$ and hyperbolic functions:

1. Theorem 1 (Even arguments):
   For $k \ge 1$:
   $$\zeta^2(2k) + \zeta(4k) = \sum_{n=1}^\infty \frac{a_k(n)}{n^{4k-1}}$$
   where $a_k(n)$ involves a sum of $\cot(\pi \varepsilon_r n)$. A corollary recovers the famous Cauchy-Lerch-Ramanujan formula for $\zeta(3)$ as a special case ($k=1$).

2. Theorem 2 (Odd arguments via mixed parity):
   For $1 \le \ell \le 2k-2$:
   $$\zeta^2(2k-\ell) = \sum_{n=1}^\infty \frac{b_{k,\ell}(n)}{n^{4k-2\ell-1}}$$
   where $b_{k,\ell}(n)$ involves the Digamma function $\psi(n\varepsilon_r)$.

   • Corollary: An explicit identity for $\zeta^2(3)$ is derived involving $\text{Im}(\psi(n\varepsilon_0) + \psi(-n\varepsilon_0))$ and a rapidly convergent series.

3. Theorem 3 (Odd arguments):
   For $k \ge 1$:
   $$\frac{1}{2}\zeta^2(2k+1) + \zeta(4k+2) = \sum_{n=1}^\infty \frac{c_k(n)}{n^{4k+1}}$$
   where $c_k(n)$ involves $\psi(-n\omega_r)$.

B. Generalized Identities for Dirichlet Series

Theorem 4, 5, and 6 extend these results to general Dirichlet series $L(s; f)$. For example, Theorem 4 states:
$$L^2(2k; f) = \sum_{m=1}^\infty \frac{g(m)}{m} \left( \sum_{n=1}^\infty \frac{f(n)}{n^{4k-1}} a_k\left(\frac{n}{m}\right) - m L(4k; f) \right)$$
This framework generates a vast array of specific identities by choosing different arithmetic functions $g(m)$ (e.g., Möbius $\mu$, divisor function $\tau$, Liouville $\lambda$, Ramanujan sums $c_m(a)$).

C. Asymptotic Behavior of Coefficients

Section 5 investigates the limit behavior of the coefficients $a_k(w)$, $b_{k,\ell}(w)$, and $c_k(w)$ as $k \to \infty$. Using the Euler-Maclaurin summation formula, the author proves that these coefficients converge to functions involving Bernoulli polynomials and simple rational functions of $w$.

4. Significance

• Elementary Nature: The proofs rely on basic algebraic identities, partial fractions, and standard properties of the Digamma function, avoiding the heavy machinery of complex analysis (residue calculus) often required for such problems.
• Unification: The paper unifies disparate identities found in Ramanujan's notebooks and subsequent literature under a single general lemma (Lemma 1).
• Generality: The extension to Dirichlet convolutions (Theorems 4-6) provides a powerful tool for generating new identities for squares of $L$-functions and other arithmetic series.
• Explicit Formulas: The paper provides explicit, rapidly convergent series for $\zeta^2(3)$ and other values, which are computationally useful and theoretically insightful regarding the structure of zeta values.
• Recovery of Classical Results: The work successfully recovers the Cauchy-Lerch-Ramanujan identity for $\zeta(3)$ and extends it to higher orders, demonstrating the robustness of the proposed method.

In summary, Korolev provides a rigorous, elementary, and highly generalizable framework for expressing squares of the Riemann zeta function (and related Dirichlet series) as series involving special functions, offering both new theoretical insights and practical computational formulas.