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On the sign changes of $ψ(x)-x$

This paper improves the lower bound for the number of sign changes of the error term $\psi(x)-x$ in the Prime Number Theorem by establishing that $\liminf_{T\to\infty} V(T)/\log T \geq \gamma_0/\pi + 1/60$ , utilizing a new density estimate for zeros of the associated $k$ -function that is over $4 \cdot 10^{21}$ times more precise than previous estimates.

Original authors: Maciej Grześkowiak, Jerzy Kaczorowski, Łukasz Pańkowski, Maciej Radziejewski

Published 2026-03-17

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CC BY 4.0

Original authors: Maciej Grześkowiak, Jerzy Kaczorowski, Łukasz Pańkowski, Maciej Radziejewski

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

The Big Picture: A Tug-of-War with Prime Numbers

Imagine you are trying to predict the weather. You have a perfect mathematical model that says, "It will rain exactly 50% of the time." But when you look at the actual weather data, it's messy. Sometimes it rains 55% of the time, sometimes 45%. It wobbles around your prediction.

In the world of mathematics, Prime Numbers are the "weather." They are the building blocks of all numbers (2, 3, 5, 7, 11...). There is a famous rule called the Prime Number Theorem that gives us a very good estimate of how many primes exist up to a certain number $x$ . Let's call this estimate $x$ .

However, the actual count of primes, denoted as $\psi(x)$ , doesn't match the estimate perfectly. It wobbles above and below the line.

When $\psi(x) > x$ , the actual count is higher than expected.
When $\psi(x) < x$ , the actual count is lower.

The paper asks a simple but deep question: How many times does this count switch from being "too high" to "too low" (or vice versa) as we look at bigger and bigger numbers?

The authors prove that this switching happens more often than anyone previously thought.

The Problem: Counting the "Switches"

Let's call the number of times the count switches sides $V(T)$ .

If you look at numbers up to 1 million, maybe it switches 10 times.
If you look up to 1 trillion, it switches more.

The paper is about proving a lower bound. They want to say: "No matter how far out you go, the number of switches is at least this much."

For decades, mathematicians had a "glass ceiling" on how high they could push this lower bound. They knew it was at least some number, but they couldn't prove it was higher. This paper smashes through that ceiling.

The Secret Weapon: The "Ghost" Zeros

To understand why the count wobbles, mathematicians look at something called the Riemann Zeta Function. This is a complex mathematical object that acts like a "frequency analyzer" for prime numbers.

Inside this function, there are special points called zeros (where the function equals zero). These zeros are like the "notes" in a musical chord that creates the wobble in the prime numbers.

The lowest "note" (the first zero) has a specific height, called $\gamma_0$ (roughly 14.13).
The paper proves that the frequency of the wobbles is directly tied to this lowest note.

The authors show that the number of switches is roughly proportional to $\frac{\gamma_0}{\pi}$ . But they found a way to add a tiny, extra "bonus" to this number, proving the switches happen even more frequently than the basic formula suggested.

How They Did It: The "Tiling" Method

This is the most creative part of the paper. The authors had to prove that there are enough "shifts" (moments in time) where the wobble crosses the line.

Imagine you have a giant, multi-dimensional room (a hypercube). You want to fill this room with tiles.

The Old Way: Previous mathematicians tried to fit tiles in the room using a rigid, grid-like pattern. They could only guarantee that a small percentage of the room was covered. It was like trying to fill a swimming pool with a few scattered bricks.
The New Way (Tiling Method): The authors developed a new technique called the "Tiling Method."
- Imagine you have a very flexible, stretchy fabric.
- Instead of placing bricks one by one, they found a way to stretch this fabric so it covers almost the entire room, leaving only tiny gaps near the walls.
- They used a computer algorithm (a version of the famous LLL algorithm) to find the perfect "stretch" that fits the room.

The Analogy:
Think of the "shifts" as people trying to find a seat in a crowded theater.

Old method: You tell people, "Sit in these specific rows." Many seats remain empty because the rows don't match the crowd's movement.
New method: You realize the crowd moves in a specific rhythm. You tell them, "Sit wherever you can fit, as long as you follow this rhythm." Suddenly, almost every seat is filled.

By filling the "room" of possibilities much more efficiently, they proved that the "switches" in the prime number count must happen more often.

The Result: A New Record

The paper concludes with a specific number. They proved that the number of switches ( $V(T)$ ) divided by the logarithm of the range ( $\log T$ ) is at least:

$\frac{\gamma_0}{\pi} + \frac{1}{60}$

The $\frac{1}{60}$ part is the "bonus." It seems small, but in the world of prime numbers, this is a massive improvement. It's like improving a GPS estimate from "You are within 10 miles" to "You are within 0.1 miles."

Why Does This Matter?

It's a Victory for Logic: It shows that even after 100 years of studying prime numbers, we can still find new, deeper truths about how they behave.
It's a Technical Masterpiece: The authors didn't just do math on a chalkboard; they combined deep theory with massive computer calculations (using "ball arithmetic" to ensure every decimal was perfect) to prove their point.
It Honors a Legend: The paper is dedicated to Alberto Perelli on his 70th birthday, celebrating a lifetime of work in this field.

Summary in One Sentence

The authors used a clever new "tiling" strategy and powerful computers to prove that the count of prime numbers wobbles back and forth across its predicted average much more frequently than we ever knew before.

1. Problem Statement

The paper addresses the problem of estimating the lower bound for $V(T)$ , the number of sign changes of the error term in the Prime Number Theorem, defined as $\psi(x) - x$ , within the interval $[1, T]$ . Specifically, the authors aim to improve the known lower bound for the limit inferior:
$\kappa_0 := \liminf_{T \to \infty} \frac{V(T)}{\log T}$
Previous results had established that $\kappa_0 \ge \frac{\gamma_0}{\pi} + \epsilon$ , where $\gamma_0 \approx 14.1347$ is the imaginary part of the first non-trivial zero of the Riemann zeta-function ( $\zeta(s)$ ). The goal was to significantly increase the additive constant $\epsilon$ , which had previously been extremely small (on the order of $10^{-30}$ ).

2. Methodology

The proof relies on a sophisticated interplay between analytic number theory, the theory of almost periodic functions, and computational verification.

A. Connection to Zeros of Almost Periodic Functions

The authors utilize a result by Kaczorowski (2011) linking $\kappa_0$ to the density of zeros of a normalized $k$ -function (an almost periodic function) on the upper half-plane:
$F^*(z) = 1 + \sum_{n=1}^{\infty} \frac{\rho_0}{\rho_n} e^{i(\gamma_n - \gamma_0)z}$
where $\rho_n = \frac{1}{2} + i\gamma_n$ are the non-trivial zeros of $\zeta(s)$ .
If the Riemann Hypothesis (RH) is true, the problem reduces to finding a lower bound for the density $\kappa$ of zeros of $F^*(z)$ . The main theorem states:
$\kappa_0 \ge \frac{\gamma_0}{\pi} + 2\kappa$
Thus, proving a larger $\kappa$ directly improves the bound for $\kappa_0$ .

B. The "Tiling Method" and Diophantine Approximation

To estimate the density of zeros, the authors employ a new "tiling method" to count "appropriate shifts" $\tau$ that map a zero of a partial sum $F_N(z)$ to a zero of the full function $F(z)$ .

Partial Sums and Contours: They consider the partial sum $F_N(z)$ (with $N=21$ ) and a specific closed contour $z_1(t)$ in the complex plane.
Proposition 1 (Sufficient Condition): They introduce a flexible condition for a shifted contour $z_1(t) + \tau$ to contain a zero. Unlike previous methods requiring uniform bounds on the distance of $\frac{\omega_n \tau}{2\pi}$ to integers, this proposition uses a weighted sum involving non-decreasing functions $w_n$ . This allows for a more efficient estimation of the valid shift set.
Density Estimation (Theorem 3): The core innovation is Theorem 3, which replaces the classical Minkowski convex body theorem estimate ( $2^{-N} \text{vol}(A)$ $2^{- N} vol (A)$ ) with a volume estimate of a subset $A' \subset A$ $A^{'} \subset A$ .
- The method involves constructing a dense lattice of reachable points using the LLL algorithm (Lenstra–Lenstra–Lovász).
- By "tiling" the solution space with these lattice points, they show that the density of valid shifts approaches the conjectural volume $\text{vol}(A)$ much more closely than previous bounds, effectively bypassing the linear independence conjecture limitations.

C. Computational Verification

The theoretical framework relies on rigorous numerical computations performed in Mathematica using the ARB library (ball arithmetic) to ensure error bounds are controlled.

Parameters: They use $N=21$ (the first 21 non-trivial zeros).
Contour Design: A specific piecewise-linear contour $z_1(t)$ and phase function $\alpha(t)$ are constructed to maximize the argument variation of the function along the path.
Verification: The authors rigorously verify that the real part of the function along the contour remains positive (satisfying the conditions of Proposition 1) and that the Diophantine approximation coefficients satisfy the required bounds.

3. Key Contributions

New Density Estimate: The paper provides a density estimate for zeros of associated $k$ -functions that is over $4 \cdot 10^{21}$ times better than previous estimates of this type.
The Tiling Method: A novel technique for estimating the density of Diophantine shifts using lattice tiling, which allows for non-convex solution sets and significantly reduces the "loss" near the boundaries of the solution space.
Flexible Sufficient Condition (Proposition 1): A generalized condition for the existence of zeros in shifted contours that utilizes weighted sums rather than uniform norms, allowing for tighter constraints on the shift parameters.
Rigorous Machine-Checked Proof: The proof incorporates extensive computer-assisted verification with rigorous error bounds, ensuring the mathematical validity of the numerical results.

4. Main Results

The authors prove Theorem 1:
$\liminf_{T \to \infty} \frac{V(T)}{\log T} \ge \frac{\gamma_0}{\pi} + \frac{1}{60}$
where $\gamma_0 \approx 14.1347$ .

Comparison to Previous Work:
- Ingham (1936): Proved positivity (conditional).
- Kaczorowski (1980s): $\kappa_0 \ge \gamma_0/\pi$ .
- Morrill, Platt, Trudgian (2023): $\kappa_0 \ge \gamma_0/\pi + 1.867 \cdot 10^{-30}$ .
- This Paper: $\kappa_0 \ge \gamma_0/\pi + 0.0166\dots$ (since $1/60 \approx 0.0166$ ).
- This represents a massive improvement, increasing the additive constant by roughly 28 orders of magnitude compared to the previous best published result.

5. Significance

Breaking the Barrier: The result breaks the long-standing barrier of $\gamma_0/\pi$ by a substantial margin, demonstrating that the oscillations of $\psi(x) - x$ are significantly more frequent than previously proven.
Methodological Advancement: The "tiling method" combined with the LLL algorithm offers a powerful new tool for problems involving the distribution of zeros of almost periodic functions and Diophantine approximation, potentially applicable to other problems in analytic number theory.
Computational Rigor: The paper sets a high standard for combining theoretical analysis with rigorous computer-assisted proofs, utilizing interval arithmetic to eliminate numerical uncertainty.
Conditional on RH: While the main result assumes the Riemann Hypothesis (to utilize the specific form of the zeros), the authors note that if RH is false, the lower bound is actually much stronger ( $\kappa_0 \ge 3 \cdot 10^{12}/\pi$ ), making the RH case the "hardest" and most significant scenario addressed.

In summary, this paper represents a landmark improvement in understanding the oscillatory behavior of the prime counting function error term, achieved through a novel synthesis of analytic techniques and high-precision computational number theory.