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Piatetski-Shapiro Primes in short intervals

This paper establishes the existence of Piatetski-Shapiro primes of the form $\lfloor n^c \rfloor$ within short intervals $[x, x + x^\theta]$ by proving both an asymptotic formula and a lower bound under specific restrictions on the parameters $\theta$ and $c$ .

Original authors: Lingyu Guo, Victor Zhenyu Guo

Published 2026-06-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Lingyu Guo, Victor Zhenyu Guo

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: Hunting for Rare Gems in a Tiny Box

Imagine you are a treasure hunter looking for primes. In the world of numbers, primes are like rare, special gems (numbers like 2, 3, 5, 7, 11 that can only be divided by 1 and themselves).

Usually, mathematicians look for these gems over a very long stretch of the number line. They know that if you look at a huge interval, you will definitely find some gems. But the big challenge in this paper is looking for gems in a very short interval.

Think of it like this: If you have a long highway (the number line), finding a gas station (a prime) is easy. But what if you are only allowed to look at a 100-meter stretch of that highway? Can you guarantee there is a gas station there? The authors of this paper are trying to prove that yes, you can find these special gems even in very short stretches, provided you know exactly what kind of gems you are looking for.

The Special Gems: "Piatetski-Shapiro" Primes

The authors aren't looking for just any prime. They are looking for Piatetski-Shapiro primes.

Imagine you have a machine that takes a whole number $n$ (1, 2, 3...) and squashes it down using a special formula: $\lfloor n^c \rfloor$ .

The symbol $\lfloor \dots \rfloor$ means "round down to the nearest whole number."
The letter $c$ is a dial you can turn. If you turn the dial to $c=2$ , you get $n^2$ (1, 4, 9, 16...). If you turn it slightly differently, you get a different sequence.

The question is: Does this squashed sequence contain infinitely many primes?
For example, if you set the dial to $c=1.5$ , do you get primes like $\lfloor 1^{1.5} \rfloor$ , $\lfloor 2^{1.5} \rfloor$ , $\lfloor 3^{1.5} \rfloor$ ... and are any of those results prime numbers?

This is a hard problem. It's like asking if a specific, weirdly shaped sieve will catch any gold dust. The authors prove that for certain settings of the dial ( $c$ ), these special primes do exist, even when you only look at a tiny, short interval of numbers.

The Two Main Results: The "Exact Map" and the "Guaranteed Treasure"

The paper provides two different ways to prove these primes exist.

1. The Exact Map (The Asymptotic Formula)

Theorem 1.1 is like drawing a precise map.

What it does: It doesn't just say "there is a prime here." It says, "If you look in this short interval, there will be exactly this many primes, give or take a tiny bit of error."
The Catch: This map only works if the "short interval" isn't too short and the "dial" ( $c$ ) isn't too weird. The authors spent a lot of time calculating the exact boundaries of where this map is valid. They found that if you adjust the length of your search interval and the setting of the dial just right, the math works out perfectly.

2. The Guaranteed Treasure (The Lower Bound)

Theorem 1.2 is a bit more flexible. It uses a tool called the Harman Sieve.

The Analogy: Imagine you have a fishing net (the sieve). You want to catch fish (primes). The "Exact Map" tells you exactly how many fish are in the net. The "Guaranteed Treasure" approach just wants to prove that the net won't be empty.
The Result: By using this sieve, the authors can prove that there is at least one prime in the interval, even in situations where the "Exact Map" is too complicated to draw. This allows them to find these special primes in a wider range of settings than the first method.

How They Did It: The "Mathematical Noise" Problem

To prove these things, the authors had to deal with Exponential Sums.

The Metaphor: Imagine you are trying to hear a single, quiet whisper (the prime numbers) in a very loud, chaotic room full of people shouting (the other numbers).
The Problem: The "shouting" is so loud and complex that it drowns out the whisper. In math, this is called "noise."
The Solution: The authors developed new techniques to filter out the noise. They used a method called the Cauchy-Schwarz inequality (a way of measuring how much two things overlap) and the A-process (a way of smoothing out the chaos).
The Short Interval Twist: Usually, mathematicians can ignore the fact that they are looking at a "short" interval because the noise averages out over long distances. But because the authors are looking at a short interval, the noise doesn't average out nicely. They had to invent a new way to listen to the whisper specifically in that tiny, noisy room.

The "Weird" Curve

The paper mentions a strange phenomenon in their results (Figure 1). As they change the length of the interval, the "dial" setting required to find primes sometimes goes up and down in a weird way.

Why? It's because of a specific mathematical term they had to ignore to keep the proof simple. They decided not to chase that specific weird curve because it only happens when the interval is already quite long, and their main goal was to find primes in the shortest possible intervals.

Summary

In short, Lingyu Guo and Victor Zhenyu Guo have proven that if you look at a short stretch of numbers and apply a specific rounding formula, you will find prime numbers inside it. They did this by:

Creating a precise count of how many primes to expect (Theorem 1.1).
Using a "sieve" to guarantee at least one prime exists (Theorem 1.2).
Developing new ways to filter out mathematical "noise" to hear the primes in a short, crowded interval.

They didn't invent a new machine or cure a disease; they simply solved a very specific, difficult puzzle about where numbers hide in the vastness of mathematics.

Technical Summary: Piatetski-Shapiro Primes in Short Intervals

Problem Statement
The paper addresses the problem of establishing the existence of Piatetski-Shapiro primes within short intervals. Specifically, it investigates the counting function $\pi_c(x)$ , which counts primes $p \le x$ of the form $p = \lfloor n^c \rfloor$ for a fixed real number $c > 1$ . The central question is whether the asymptotic relation or a lower bound for the difference $\pi_c(x + x^\theta) - \pi_c(x)$ holds for specific ranges of the short interval parameter $\theta \in (2/3, 1)$ and the exponent parameter $\gamma = c^{-1}$ . This problem serves as an approximation to the broader conjecture regarding the infinitude of primes of the form $n^2 + 1$ . While the existence of such primes in long intervals is well-established (with the admissible range for $c$ extended to $c < 2817/2426$ by Rivat and Sargos), the behavior in short intervals $[x, x + x^\theta]$ remains a core challenge in analytic number theory.

Methodology
The authors employ a combination of exponential sum estimation and sieve methods to derive their results. The core technical machinery involves:

Exponential Sum Bounds: The proof relies heavily on estimating exponential sums of the form:
$\sum_{h \sim H} \delta_h \sum_{n \sim N} \sum_{m} \mathbb{1}_{mn \in I} a_n b_m e(h(mn)^\gamma)$
where $I = (x, x + x^\theta]$ . The authors distinguish between Type I sums (where coefficients are simple, e.g., $b_m=1$ ) and Type II sums (where coefficients are more complex, e.g., $|b_m| \ll x^\epsilon$ ).
- Type I: Unlike standard approaches using double large sieves, the authors estimate the inner sum over $m$ first using Lemma 2.5 (a higher-order derivative test) to handle the entanglement of $m$ and $n$ caused by the short interval condition $mn \in I$ .
- Type II: Two distinct methods are utilized. The first applies the Cauchy-Schwarz inequality and the $A$ -process (Weyl-van der Corput inequality) to convert the sum into a form amenable to short interval information. The second method adapts Heath-Brown's technique of counting rational points to extract additional Type II information.
Harman Sieve: To obtain lower bound results (Theorem 1.2), the authors apply the Harman sieve. This involves decomposing the counting function using the Buchstab identity and the Fundamental Lemma (Lemma 5.3). The Fundamental Lemma allows the authors to transfer asymptotic formulas from the set of integers in the interval to the set of Piatetski-Shapiro integers, provided the exponential sum bounds hold.
Optimization: The ranges for $\theta$ and $\gamma$ are determined by optimizing the parameters in the exponential sum estimates and the sieve weights. The authors utilize numerical integration (via MATLAB) to calculate the contributions of discarded terms in the sieve process, specifically involving Buchstab's function $w(u)$ .

Key Contributions and Results

The paper presents two main theorems establishing the existence of Piatetski-Shapiro primes in short intervals under specific constraints on $\theta$ and $\gamma$ :

Theorem 1.1 (Asymptotic Formula): The authors prove an asymptotic formula for $\pi_c(x + x^\theta) - \pi_c(x)$ of the form $\gamma x^{\theta+\gamma-1}/\log x$ . This holds for $\theta \in (2/3, 1)$ and $\gamma$ satisfying a piecewise-defined lower bound (Equation 1.2). The admissible range for $\gamma$ varies with $\theta$ , generally requiring $\gamma$ to be close to 1 for smaller $\theta$ . Notably, the paper observes an unusual phenomenon where the admissible $\gamma$ increases with $\theta$ near $\theta \approx 0.9$ , attributed to the term $(1+\theta)/2$ in the Type II sum estimation.
Theorem 1.2 (Lower Bound): By employing the Harman sieve, the authors improve the admissible range for $\gamma$ compared to Theorem 1.1. For fixed values of $\theta \in (2/3, 1)$ , they establish a lower bound $\pi_c(x + x^\theta) - \pi_c(x) \gg x^{\theta+\gamma-1}/\log x$ . Table 1 provides specific values of $\gamma_0$ such that the result holds for all $\gamma > \gamma_0$ . For instance, at $\theta = 0.90$ , the result holds for $\gamma > 0.9168$ , whereas Theorem 1.1 only guarantees an asymptotic formula for $\gamma > 0.9534$ .

Significance and Claims
The paper claims to provide the first results on Piatetski-Shapiro primes in short intervals with both asymptotic formulas and improved lower bounds. The significance lies in extending the known admissible ranges for the exponent $c$ (via $\gamma$ ) in the context of short intervals, which is a more difficult setting than long intervals due to the lack of cancellation in the exponential sums.

The authors explicitly note that their method for Type I sums is tailored to the short interval constraint, as standard double large sieve techniques are less effective when $\theta$ is small. They also acknowledge that while they could potentially improve the range for larger $\gamma$ (specifically $\gamma > 0.87$ ) with more careful calculations, their primary focus is on the case where $\theta$ is small. The results are presented as a direct application of established sieve techniques and exponential sum bounds to the specific structure of Piatetski-Shapiro sequences, without claiming to solve the $n^2+1$ conjecture or proposing new experimental applications.