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A new expansion of the Riemann zeta function

This paper introduces a convergent series expansion of the Riemann zeta function within the critical strip, utilizing polynomials with roots on the critical line and coefficients derived from the Euler constant and even zeta values, following Ramanujan's method of summation.

Original authors: B. Candelpergher

Published 2026-03-03

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: B. Candelpergher

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Riemann Zeta function ( $\zeta(s)$ ) as a mysterious, infinitely complex musical instrument. For over a century, mathematicians have been trying to understand its "sound," especially in a specific, dangerous region called the Critical Strip (where the real part of the number $s$ is between 0 and 1). This is the "Goldilocks zone" where the famous Riemann Hypothesis lives—the idea that all the instrument's most important notes (its "roots" or zeros) are perfectly tuned to a single, straight line.

In this paper, author B. Candelpergher introduces a new way to play this instrument. Instead of trying to describe the sound with a chaotic jumble of notes, he builds a new set of musical scales (orthogonal polynomials) that fit the instrument perfectly.

Here is the breakdown of the paper's journey, translated into everyday language:

1. The Problem: A Messy Orchestra

Traditionally, mathematicians describe the Zeta function using infinite sums that are often messy or divergent (they go on forever without settling down). It's like trying to describe a symphony by listing every single sound wave in the air at once. It's accurate, but it's hard to work with.

Candelpergher asks: Can we break this complex sound down into a neat, organized series of building blocks?

2. The Solution: A New Set of Lego Bricks

The author invents a family of special mathematical shapes called polynomials (let's call them $P_m$ ). Think of these as a custom set of Lego bricks.

The Magic Property: The most exciting thing about these bricks is that if you look at where they "break" (their roots), they all line up perfectly on a single vertical line.
The Connection: This line is exactly the Critical Line ( $\Re(s) = 1/2$ ) where the Riemann Hypothesis says the Zeta function's zeros live.
The Analogy: It's as if the author discovered that the Zeta function is made entirely of bricks that naturally align with the very line mathematicians have been hunting for. This suggests a deep, hidden harmony in the structure of numbers.

3. The Secret Ingredient: Ramanujan's "Magic Sum"

To build this new series, the author uses a tool called Ramanujan's Summation.

The Problem: Sometimes you have a list of numbers that adds up to infinity (like $1 + 2 + 3 + 4 + \dots$ ). In normal math, this is "broken."
The Magic Trick: The Indian genius Srinivasa Ramanujan found a way to assign a finite, meaningful value to these broken sums. It's like taking a chaotic, infinite crowd and asking them to sit down in a specific order so you can count them.
The Result: By using this "magic counting" method, the author can turn a divergent, messy series into a clean, convergent one. He essentially says, "If we look at this infinite list through Ramanujan's special lens, it actually makes perfect sense."

4. The New Formula: A Clean Recipe

The paper's main achievement is a new formula:
$\zeta(s) = \text{Sum of (Coefficients)} \times \text{(Our Special Bricks)}$

The Coefficients: These are the "weights" or "amounts" of each brick you need. Surprisingly, these weights are made of familiar constants like Euler's constant ( $\gamma$ ) and values of the Zeta function itself ( $\zeta(2), \zeta(3)$ , etc.).
The Convergence: Unlike older methods, this new series actually stops growing and settles down to the correct answer within the Critical Strip. It's a stable, reliable recipe.

5. The Grand Finale: A Perfect Harmony

The paper ends with a beautiful observation about the "energy" of the Zeta function.

The author treats the Zeta function on the critical line as a wave.
Because his new "bricks" (the polynomials) are orthogonal (they are like perfectly perpendicular axes on a graph, or notes that don't interfere with each other), he can calculate the total "volume" or energy of the Zeta function.
The Result: He derives a specific number for the total energy of the Zeta function on that critical line: $\log(2\pi) - \gamma - 1/2$ .

Why Does This Matter?

Think of the Riemann Zeta function as a locked safe containing the secrets of prime numbers. For a long time, we had a key that was rusty and hard to turn.

Candelpergher has forged a new, smooth key.

It fits the lock perfectly (the roots align with the critical line).
It turns easily (the series converges nicely).
It reveals the inner mechanism (the coefficients are built from fundamental constants).

While this doesn't immediately prove the Riemann Hypothesis, it provides a powerful new map and a new set of tools for mathematicians to explore the most mysterious region of number theory. It suggests that the chaotic distribution of prime numbers might be governed by a very orderly, orthogonal structure that we are just beginning to understand.

1. Problem Statement

The paper addresses the representation of the Riemann zeta function, $\zeta(s)$ , within the critical strip ( $0 < \Re(s) < 1$ ). While $\zeta(s)$ is well-understood via its Dirichlet series (convergent for $\Re(s) > 1$ ) and its functional equation, finding convergent series expansions in the critical strip that utilize orthogonal polynomials and explicitly relate to fundamental constants (like the Euler-Mascheroni constant $\gamma$ and values of $\zeta(k)$ ) is a significant challenge.

Existing polynomial expansions often rely on Newton interpolation or Stieltjes constants. This paper aims to introduce a new family of orthogonal polynomials $P_m(s)$ whose roots lie strictly on the critical line $\Re(s) = 1/2$ , and to express $\zeta(s)$ as a convergent series of these polynomials with coefficients derived from rational combinations of $\gamma$ and $\zeta(k)$ .

2. Methodology

The author employs a synthesis of several advanced mathematical tools:

Laguerre Polynomials and Mellin Transforms: The core of the method involves the sequence of Laguerre polynomials $L_m(x)$ . The author defines a new sequence of polynomials $P_m(s)$ via the Mellin transform of the function $x \mapsto e^{-x}L_m(2x)$ .
Generating Functions: A generating function relation is established linking $z^{-s}$ to a series involving $P_m(s)$ and a specific kernel $\psi_m(z) = \frac{1}{1+z}(\frac{z-1}{z+1})^m$ .
Ramanujan's Summation: Since the direct summation of the series derived from the generating function leads to divergent series in the critical strip, the author utilizes Ramanujan's summation ( $\mathcal{R}\sum$ ). This technique assigns finite values to divergent series by using analytic continuation and the Plana summation formula.
Orthogonal Systems: The paper utilizes the Mellin-Parseval formula to establish that the polynomials $Q_m(t) = \frac{1}{2}P_m(\frac{1}{2} + it)$ form a complete orthonormal system in the weighted $L^2$ space $L^2(\mathbb{R}, \frac{dt}{\cosh(\pi t)})$ .

3. Key Contributions and Results

A. Definition and Properties of $P_m(s)$

The polynomials $P_m(s)$ are defined by the integral:
$\Gamma(s)P_m(s) = 2 \int_0^{+\infty} e^{-x}L_m(2x)x^{s-1} dx$
They can also be expressed explicitly as:
$P_m(s) = \sum_{k=0}^m \binom{m}{k} (-1)^k 2^{k+1} \frac{(s)_k}{k!}$
where $(s)_k$ is the Pochhammer symbol.

Roots: A major result is the proof that all roots of $P_m(s)$ lie on the critical line $\Re(s) = 1/2$ . This is a consequence of the fact that the transformed polynomials $Q_m(t)$ are orthogonal with respect to the weight $1/\cosh(\pi t)$ .

B. The Expansion of $\zeta(s)$

The paper derives the following convergent expansion for $0 < \Re(s) < 1$ :
$\zeta(s) = \sum_{m=0}^{\infty} x_m P_m(s)$
The coefficients $x_m$ are explicitly calculated as linear combinations of the Euler constant $\gamma$ and the values $\zeta(2), \zeta(3), \dots, \zeta(m+1)$ .
The formula for $x_m$ is:
$x_m = \gamma + \sum_{k=1}^m \binom{m}{k} (-2)^k \zeta(k+1) + r_m$
where $r_m$ is a rational number given by:
$r_m = (-1)^{m+1} + \sum_{k=1}^m \frac{1 + (-1)^{k-1}}{k}$
Specific examples provided include:

$x_1 = \gamma - 2\zeta(2) + 3$
$x_2 = \gamma - 4\zeta(2) + 4\zeta(3) + 1$

C. Ramanujan Summation of Divergent Series

The derivation relies heavily on evaluating the Ramanujan sum of the series:
$z_m = \mathcal{R}\sum_{n \ge 1} \frac{1}{n+1} \left(\frac{n-1}{n+1}\right)^m$
The author proves that for $s \neq 1$ , $\mathcal{R}\sum_{n \ge 1} n^{-s} = \zeta(s) - \frac{1}{s-1}$ , allowing the transition from the divergent series to the analytic function $\zeta(s)$ .

D. Orthonormal Expansion and Integral Identities

The expansion is shown to be an orthonormal expansion in the space $L^2(\mathbb{R}, \frac{dt}{\cosh(\pi t)})$ . Specifically:
$\zeta\left(\frac{1}{2} + it\right) = 2 \sum_{m=0}^{\infty} x_m Q_m(t)$
This leads to a Parseval-type identity for the squared modulus of the zeta function on the critical line:
$\frac{1}{2} \int_{-\infty}^{+\infty} \left| \zeta\left(\frac{1}{2} + it\right) \right|^2 \frac{dt}{\cosh(\pi t)} = \log(2\pi) - \gamma - \frac{1}{2}$
This integral identity is derived formally using the Laplace transform of the expansion and Ramanujan summation techniques.

4. Significance

Geometric Insight: The construction of polynomials $P_m(s)$ with roots exactly on the critical line provides a new algebraic structure associated with the zeta function, potentially offering new perspectives on the distribution of zeta zeros.
Computational Utility: The expansion expresses $\zeta(s)$ in the critical strip using a rapidly converging series of polynomials with coefficients involving standard constants ( $\gamma, \zeta(k)$ ). This could be useful for high-precision numerical computations of $\zeta(s)$ in the critical strip.
Unification of Concepts: The paper elegantly bridges Laguerre polynomials, Mellin transforms, Ramanujan's summation of divergent series, and the theory of orthogonal polynomials on the critical line.
New Integral Formulas: The derived integral identity for $\int |\zeta(1/2+it)|^2 \frac{dt}{\cosh(\pi t)}$ provides a new global constraint on the behavior of the zeta function on the critical line, linking it directly to $\log(2\pi)$ and $\gamma$ .

In summary, Candelpergher presents a novel, rigorous, and convergent polynomial expansion of the Riemann zeta function in the critical strip, grounded in the theory of orthogonal polynomials and Ramanujan summation, yielding both explicit coefficient formulas and new integral identities.