🔢 mathematics

Exponential sums over primes are unbounded

This paper proves that exponential sums over primes exhibit no better than square root cancellation on average over short intervals, thereby establishing a lower bound of $x^{1/6 - \varepsilon}$ for their supremum and resolving a question posed by Aymone and Ramaré.

Original authors: Péa Bazin

Published 2026-08-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Péa Bazin

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 you are a detective trying to solve a mystery about how numbers are arranged. In the world of mathematics, there is a special club called "Number Theory," where experts study the hidden patterns of whole numbers. One of the most famous members of this club is the "prime number." Primes are the building blocks of all numbers (like 2, 3, 5, 7, 11), and they seem to appear in a completely random, chaotic way. Because they are so unpredictable, mathematicians often try to "tame" them by adding them up in specific ways, a process called an "exponential sum." Think of this like trying to predict the weather by adding up the temperature, humidity, and wind speed every day. Sometimes, if you add up enough random numbers, the chaos cancels itself out, and you get a result that is very close to zero. This is called "cancellation."

For a long time, mathematicians wondered if these prime number sums could ever get huge and keep growing forever, or if they would always stay small and manageable. It's like asking if a chaotic crowd of people will eventually settle down into a quiet line, or if they will keep running around wildly. The question is: no matter how you look at the primes, is there always a moment where the "noise" gets so loud it breaks the silence? This isn't just a game; understanding these patterns helps us understand the very fabric of numbers, which is crucial for things like secure computer encryption.

In this paper, a mathematician named Pierre-Alexandre Bazin steps into the detective's shoes to answer a specific question: Do these prime sums ever get unbounded? In other words, can they grow as large as we want? The answer, according to Bazin, is a resounding "yes." He proves that these sums do not just stay small; they can get very large, at least on average, when you look at them over short stretches of numbers.

To understand how he did this, imagine you are listening to a radio that is full of static. Usually, if you tune to a specific station, the static cancels out, and you hear a clear song. But Bazin shows that if you listen to the primes on a very short "interval" (a tiny slice of time or a small range of numbers), the static doesn't cancel out as nicely as we hoped. Instead of the noise disappearing, it stays loud. He proves that the "volume" of this noise is at least as big as the square root of the number of primes you are counting. This is a big deal because it means the primes are even more stubborn and chaotic than some previous theories suggested.

Bazin didn't just look at primes; he also looked at "divisors" (numbers that divide evenly into others, like how 2 and 3 divide 6). He found that for divisors, the noise is even more predictable and follows a very specific, almost perfect pattern, which helps him prove his point about the primes.

The main discovery of the paper is a proof that for any specific setting (represented by a number called $\alpha$ ), there is always a point where the sum of the primes gets surprisingly large. Specifically, he shows that the largest sum you can find is at least as big as the number of primes to the power of $1/6$ (which is like taking the cube root of the square root). While this might sound like a small number, in the world of infinite primes, it means the sums are unbounded—they can grow forever.

He also shows that this happens even if you only look at very short intervals of numbers. If you take a short list of numbers, say from 1 to 100, and then from 101 to 200, and so on, the "noise" in these short lists doesn't disappear. It stays strong. This answers a question posed by other mathematicians, Aymone and Ramaré, who were wondering if the sums could ever get out of control. Bazin proves they can.

One interesting twist is that his proof works best when the numbers are "irrational" (numbers that go on forever without repeating, like $\pi$ ). If the numbers are "rational" (simple fractions), the behavior is different and easier to predict. But for the messy, irrational numbers, the primes refuse to settle down. The paper also mentions that if we assume a famous, unproven idea called the "Generalized Riemann Hypothesis" (which is like a super-powerful rule about how numbers behave), we can push these results even further, showing the sums get large even in longer intervals.

In short, this paper is a victory for the chaotic side of mathematics. It proves that the prime numbers are so wild that no matter how you try to smooth them out with math, they will always have moments of huge, unbounded energy. It's a reminder that in the universe of numbers, chaos is not just a possibility; it's a guarantee.

Technical Summary: Exponential sums over primes are unbounded

Problem Statement
The paper addresses a question posed by Aymone and communicated by Ramaré regarding the behavior of prime exponential sums. Specifically, it investigates whether the sum
$\pi(x; \alpha) := \sum_{p \le x} e(\alpha p)$
is unbounded for all real $\alpha$ , where $e(\beta) = e^{2\pi i \beta}$ . While upper bounds for these sums are well-understood (e.g., Vinogradov's theorem provides non-trivial upper bounds depending on the Diophantine approximation properties of $\alpha$ ), non-trivial lower bounds for fixed $\alpha$ have remained elusive. The trivial $L^2$ average is known to be of order $x \log x$ , but pointwise lower bounds for fixed $\alpha$ were previously unknown beyond the $L^1$ average result of Vaughan.

Methodology
The author develops a new method to establish lower bounds for exponential sums of the form $\sum f(n)e(\alpha n)$ by analyzing their behavior on short intervals. The core strategy involves:

Short Interval Analysis: Instead of requiring the global Diophantine approximation parameter $R(x, \alpha)$ to be small (which corresponds to $\alpha$ being close to a rational with small denominator), the method requires the local parameter $R(y, \alpha)$ to be small for a short interval length $y$ . This allows the exponential sum on an interval $[n, n+y]$ to be approximated by sums at rational points $a/q$ .
Large Sieve Application: The author applies the Montgomery-Vaughan large sieve to the function $g(n) = F(n, n+y; \alpha)$ , where $F$ is the partial sum of the arithmetic function. This yields a lower bound for the average of the squared modulus of the short interval sums, $\frac{1}{x} \sum |F(n, n+y; \alpha)|^2$ , in terms of the values of the global sums at rational points $F(x; a/q)$ .
Major Arc Estimation: The lower bound depends on the existence of "major arcs" (rationals $a/q$ $a / q$ close to $\alpha$ $α$ ) within the interval of approximation.
- For the case of the divisor function $\tau$ , the author uses precise asymptotic formulas for $\sum \tau(n)e(an/q)$ to show that the main term dominates.
- For the von Mangoldt function $\Lambda$ (primes), the author relies on the Bombieri-Vinogradov theorem (via Liu and Zhan) to handle the average case where $y \ll x^{1/3}$ .
- For larger $y$ (specifically $y \gg x^{1/3}$ ), the author assumes the Generalized Riemann Hypothesis (GRH) to control the error terms for all major arcs individually, combined with a result by Heath-Brown regarding the distribution of square-free denominators in Diophantine approximations.

Key Results

Unboundedness of Prime Exponential Sums:
The paper proves that for any $\varepsilon > 0$ , the prime exponential sum is unbounded with a quantitative lower bound:
$\inf_{\alpha \in \mathbb{R}} \sup_{n \le x} |\psi(n; \alpha)| \gg x^{1/6-\varepsilon}$
where $\psi(x; \alpha) = \sum_{n \le x} \Lambda(n)e(\alpha n)$ . This implies the same result for $\pi(x; \alpha)$ (sum over primes).
Average Lower Bound on Short Intervals:
The main technical theorem (Theorem 2) establishes that for $y \le x^{1/3-\varepsilon}$ , the average of the squared sums over short intervals satisfies:
$\frac{1}{x} \sum_{-y < n \le x} \left| \sum_{n < m \le n+y} \Lambda(m)e(\alpha m) \right|^2 \gg y \log x$
This demonstrates that "square root cancellation" (the expected behavior for random sums) is the best possible on average for these intervals; the sums do not exhibit better cancellation.
Results under GRH:
Assuming the Generalized Riemann Hypothesis, the range of $y$ can be extended. For irrational $\alpha$ with suitable Diophantine approximations, the lower bound holds for $y \le x^{5/9-\varepsilon}$ , leading to the corollary:
$\limsup_{n \to \infty} n^{-5/18+\varepsilon} |\psi(n; \alpha)| > 0$
Divisor Function Case:
The method is applied to the divisor function $\tau$ . The paper shows that for irrational $\alpha$ , the sum $\sum_{n \le x} \tau(n)e(\alpha n)$ is unbounded, with a lower bound of order $x^{1/2}(\log x)(\log \log x)^{-2}$ . The author notes that this result is sharp up to a logarithmic factor for almost all $\alpha$ , based on the Carleson-Hunt inequality.

Significance and Claims
The paper claims to answer the question of Aymone and Ramaré positively, proving that prime exponential sums are unbounded for all $\alpha$ . The significance lies in:

Providing the first non-trivial lower bound for fixed $\alpha$ for prime exponential sums, moving beyond average results.
Introducing a method that shifts the requirement from global Diophantine properties ( $R(x, \alpha)$ ) to local properties ( $R(y, \alpha)$ ), allowing for the analysis of sums on short intervals.
Demonstrating that the "square root cancellation" heuristic is tight on average for short intervals involving primes, specifically showing that the sums cannot be significantly smaller than $y^{1/2}$ .

The author explicitly states that the method is general and has been applied to other arithmetic functions (such as $1_{\Omega(n)=k}$ and general multiplicative functions) in separate works, but this paper focuses on the cases of $\Lambda$ and $\tau$ . The results are presented as conditional on standard conjectures (GRH) for the extended range of $y$ , and unconditional for the range $y \le x^{1/3-\varepsilon}$ .

Technical Summary: Exponential sums over primes are unbounded

More like this