🔢 mathematics

A few remarks on the Baez-Duarte Criterion

This paper derives several significant lemmas related to the Baez-Duarte criterion.

Original authors: Alexandre Pyvovarov

Published 2026-07-15

📖 6 min read🧠 Deep dive

Original authors: Alexandre Pyvovarov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 Hypothesis as the ultimate treasure map hidden inside a giant, infinite library. For over 160 years, mathematicians have been trying to find the "X" that marks the spot, but the map is written in a code so complex that no one has cracked it yet. This paper by Alexandre Pyvovarov doesn't claim to have found the treasure, but it does hand us a very shiny, very interesting new compass and a few clever tricks for navigating the library's most confusing aisles.

The Library and the "1" Problem

The paper starts by setting up a special room in this library called a "Hilbert space." Think of this room as a giant, infinite trampoline where every possible function (a mathematical recipe for a curve) is a bouncy ball. The goal of the famous B´aez-Duarte criterion is to see if we can build a perfect, flat "1" (a constant function that never changes) by stacking and bouncing other specific balls together.

If we can build this perfect "1" out of these specific bouncy balls, then the Riemann Hypothesis is true. If we can't, it's false. The paper focuses on a specific set of balls generated by a family of functions involving the "fractional part" of numbers (the bits left over after you divide).

The Mobius Function: The Library's Chaos Agent

To build our "1," the author looks at a very strange character named the Mobius function (denoted as $\mu$ ). Imagine the Mobius function as a chaotic librarian who assigns a value of $+1$ , $-1$, or $0$ to every number based on its prime factors.

If a number is made of distinct prime factors (like $6 = 2 \times 3$ ), the Mobius function flips a coin: $+1$ or $-1$.
If a number has a repeated prime factor (like $12 = 2 \times 2 \times 3$ ), the Mobius function says "0" and ignores it completely.

The paper proves a fascinating fact: if you take all these chaotic Mobius numbers and mix them with specific "floor function" recipes, they satisfy an identity that looks like they cancel out to a perfect "1" in a pointwise sense. However, there is a catch: the paper explicitly states that if you try to use this "natural" sum of Mobius terms to build the function in the Hilbert space, the pile gets too big and unstable; it "diverges," meaning it flies off the trampoline and never settles. You cannot simply add them up directly to get the result.

The "Exponential Correction" Trick

Here is where the paper gets playful. The author tries to build that perfect "1" using the Mobius function's chaos, but since the natural sum diverges, a new strategy is needed. To fix this, the author introduces a "dampener" or a "brake" called an exponential correction. Imagine you are trying to stack a tower of blocks that keeps wobbling. The author suggests adding a special glue that gets stronger the further you go up the tower. Mathematically, this is a factor of $e^{-nu}$ (where $u$ is a small number).

The paper shows that if you use this glue, the tower becomes stable. The author then asks a critical question: "What happens if we slowly remove the glue (let $u$ get closer to zero)?"

The Good News: The paper proves that the "inner product" (a measure of how well the tower matches the target "1") is defined by a series that converges uniformly. This means the function describing this match is smooth and continuous all the way down to zero, behaving nicely as the correction vanishes.
The Hard Part: The paper does not prove that the tower itself (the total size or "norm" of the function) stays small enough to fit in the room. It shows that if the tower does stay small enough (specifically, if the limit of its size is 1), then the Riemann Hypothesis is true. But proving the tower stays small is the missing piece of the puzzle.

The Polynomial Puzzle

In the second half of the paper, the author swaps the infinite library for a set of polynomial puzzles. These are like algebraic shapes that get more complex as you add more pieces. The author defines a sequence of these shapes, $Q_k$ , which are averages of simpler shapes called $R_k$ .

The paper proves that as you add more pieces (as $k$ gets larger), these polynomial shapes get closer and closer to a specific target curve called $g(x)$ .

The Bound: The author calculates exactly how fast they get closer. The error (the distance between the shape and the target) shrinks incredibly fast, following a formula involving $e^{-c(\log k)^{3/5}/(\log \log k)^{1/5}}$ .
The Analogy: Imagine trying to draw a perfect circle using a jagged, pixelated line. As you add more pixels, the jagged line smooths out. This paper proves that the jagged line smooths out at a specific, incredibly rapid rate, but it doesn't claim the line ever becomes perfectly smooth in a way that solves the Riemann Hypothesis on its own.

What the Paper Rules Out

The paper is very careful not to claim it has solved the Riemann Hypothesis. In fact, it explicitly rules out the idea that a simple, direct sum of Mobius functions works. It shows that the "natural" way to approximate the number 1 (just adding up the Mobius terms) fails because it diverges. You must use the exponential correction (the "glue") to make it work, and even then, the final step of proving the limit exists is left as a difficult challenge.

How Sure Are We?

The author is very confident about the math they have done. They have proved several lemmas:

The Mobius function satisfies an identity with floor functions that points toward a "1," but the direct sum diverges in the Hilbert space.
The "exponential correction" creates a continuous function where the inner product with "1" behaves well (is uniformly convergent) as the correction gets smaller.
The polynomial approximations converge to the target curve at a specific, proven rate.

However, the paper does not prove the final step. It suggests that if a certain limit (the size of the tower as the glue is removed) equals 1, then the Riemann Hypothesis is true. But the paper stops short of proving that this limit is 1. It provides the tools and the map, but the final destination remains unconfirmed.

In short, this paper is like a master carpenter showing you a new, incredibly precise saw and a set of blueprints that could build a perfect chair. The carpenter proves the saw cuts perfectly and the blueprints are mathematically sound, but they haven't actually built the chair yet. They've just shown that if you follow these specific steps, the chair might be the one you've been looking for all along.

Technical Summary: A Few Remarks on the B´aez-Duarte Criterion

Problem Statement

The paper addresses the Riemann Hypothesis (RH) through the lens of the Nyman-Beurling criterion and its stronger variant, the B´aez-Duarte criterion. The core problem is to characterize the Riemann Hypothesis via the density of a specific subspace within a Hilbert space of square-integrable functions.

Specifically, let $H$ be the Hilbert space of functions $f: [1, +\infty[ \to \mathbb{C}$ such that $\int_1^\infty |f(t)|^2 t^{-2} dt < \infty$ , equipped with the scalar product $\langle f|g \rangle = \int_1^\infty f(t)\overline{g(t)}t^{-2} dt$ . Let $B_{nat}$ be the subspace generated by the family of functions $\gamma_n(t) = [\frac{t}{n}] - \frac{[t]}{n}$ for $n \in \mathbb{N} \setminus \{0\}$ . The B´aez-Duarte criterion states that the Riemann Hypothesis is equivalent to the condition that the constant function $1$ belongs to the closure of $B_{nat}$ (denoted $\overline{B_{nat}}$ ).

The paper aims to derive lemmas related to approximating the constant function $1$ using linear combinations of the basis elements $\gamma_n$ , specifically investigating the convergence properties of sequences involving the Möbius function $\mu(n)$ .

Methodology

The author employs a combination of analytic number theory, functional analysis, and asymptotic analysis. The methodology proceeds through the following steps:

Functional Setup and Integral Representations: The paper establishes the relationship between the scalar product in $H$ and the Riemann zeta function $\zeta(z)$ . It derives the identity $\langle t^{1-z} | \gamma_n \rangle = (\frac{1}{nz} - \frac{1}{n})\frac{\zeta(z)}{z}$ for $\text{Re}(z) > 1/2$ . This connects the geometry of the Hilbert space to the zeros of $\zeta(z)$ .
Möbius Inversion and Approximation: The author investigates the natural approximation of the function $1$ using the Möbius function, defined as $f_N = \sum_{n=1}^N \mu(n)\gamma_n$ . Citing B´aez-Duarte, the paper notes that this specific sequence diverges in $H$ . Consequently, the author introduces a parameterized family of approximations to regularize the divergence.
Exponential Correction: A family of functions $f(u)$ is defined for a parameter $u > 0$ :
$f(u)(t) = \sum_{n=1}^\infty \mu(n)e^{-nu}\gamma_n(t)$
The paper analyzes the continuity of the norm $\|f(u)\|_2$ and the scalar product $\langle 1 | f(u) \rangle$ as $u \to 0$ .
Asymptotic Analysis of Series: The author derives explicit formulas for the scalar products $\langle \gamma_n | \gamma_m \rangle$ involving cotangent sums and logarithmic terms, referencing prior work by Vasyunin. These are used to expand the squared norm $\|f(u)\|_2^2$ into power series involving $\mu(n)$ , $\log(n)$ , and arithmetic functions.
Polynomial Approximation and Error Bounds: The paper introduces polynomials $Q_k(x)$ related to the partial sums of the Möbius function and compares them to the function $g(x) = \sum \frac{\mu(n)}{n}x^n$ . Using the Vinogradov-Koborov zero-free region for the zeta function, the author establishes unconditional bounds on the error of these polynomial approximations.

Key Contributions and Results

1. Modified Criterion and Contradiction Argument

The paper reiterates that if $1 \in \overline{B_{nat}}$ , then $\zeta(z) \neq 0$ for $\text{Re}(z) > 1/2$ . The argument relies on the fact that if $1$ is in the closure, there exists a sequence $s_n \in B_{nat}$ converging to $1$. If a zero $z_0$ existed with $\text{Re}(z_0) > 1/2$ , the scalar product $\langle t^{1-z_0} | s_n \rangle$ would vanish, leading to the contradiction $1/z_0 = 0$ .

2. Analysis of the Exponential Correction

The paper proves that for $u > 0$ , the function $f(u)$ belongs to $B_{nat}$ .

Continuity: It is shown that the function $u \mapsto \langle 1 | f(u) \rangle = -\sum_{n=1}^\infty \frac{\mu(n)}{n} \log(n) e^{-nu}$ converges uniformly on $[0, +\infty[$ and is continuous at $u=0$ , with $\lim_{u \to 0} \langle 1 | f(u) \rangle = 1$ .
Norm Convergence Condition: The paper establishes that the Riemann Hypothesis holds if and only if $\lim_{u \to 0} \|f(u)\|_2^2 = 1$ . The difficulty lies in proving this limit, as the norm involves complex interactions of the Möbius function.

3. Power Series Expansion of the Norm

The author derives a detailed expansion for $F(x) = \|f(-\log x)\|_2^2$ for $x \in (0, 1)$ . This expansion involves products of power series of the form $\sum \mu(n)x^n$ , $\sum \frac{\mu(n)}{n}x^n$ , and $\sum \frac{\mu(n)\log n}{n}x^n$ , along with terms involving cotangent sums.

Asymptotic Cancellation: By substituting asymptotic formulas for the cotangent sums (involving $\log n$ and the Euler constant $\gamma$ ), the author shows that divergent terms cancel out, leaving a function $G(x)$ that is continuous at $x=1$ with $G(1) = -1$ .
Lower Bound: A lemma is proven stating that if the limit $L = \lim_{x \to 1} F(x)$ exists, then $L \ge 1$ .

4. Polynomial Approximations and Error Bounds

The paper defines polynomials $Q_k(x)$ as averages of partial sums related to the Möbius function.

Convergence: It is proven that $Q_k$ converges uniformly to $g(x)$ on compact subsets of $[0, 1)$ .
Unconditional Error Bound: Leveraging the Vinogradov-Koborov zero-free region, the paper derives an unconditional bound for the approximation error:
$\|Q_k - g\|_\infty \ll \exp\left(-c (\log k)^{3/5} (\log \log k)^{-1/5}\right)$
This result provides a quantitative measure of how well these polynomials approximate the limiting function without assuming the Riemann Hypothesis.

5. Continuity of the Limit Function

The paper concludes by proving that the series $\sum_{k \ge 1} \frac{\|h_k\|}{k(k+1)}$ converges, where $h_k$ are components of the approximation. This implies that the function defined by this series is continuous, reinforcing the analytic properties of the approximations used in the criterion.

Significance and Claims

The paper positions itself as a collection of "interesting lemmas" related to the B´aez-Duarte criterion rather than a proof of the Riemann Hypothesis. Its significance lies in:

Refining the Approximation Strategy: It moves beyond the divergent natural approximation ( $\sum \mu(n)\gamma_n$ ) by introducing exponential corrections and analyzing their convergence properties.
Connecting Geometry and Arithmetic: It explicitly links the norm convergence in the Hilbert space to the asymptotic behavior of series involving the Möbius function and the Riemann zeta function.
Unconditional Bounds: It provides rigorous, unconditional error bounds for polynomial approximations of the Möbius-related function $g(x)$ , utilizing the best-known zero-free regions of the zeta function.
Clarifying the Limit Condition: It isolates the specific condition ( $\lim_{u \to 0} \|f(u)\|_2^2 = 1$ ) that, if proven, would confirm the Riemann Hypothesis, thereby clarifying the precise analytical hurdle remaining in this approach.

The author remains modest, noting that while the natural approximation diverges and the exponential correction requires further analysis of the norm limit, the derived lemmas and bounds offer a structured framework for investigating the criterion.