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Real exponential sums over primes and prime gaps

This paper claims to prove that for any $0 < \lambda < 1$ , the number of primes in the short interval $(x, x + x^\lambda]$ is asymptotically $\frac{x^\lambda}{\log x}$ , thereby resolving long-standing conjectures about prime distribution such as Legendre's conjecture.

Original authors: Luan Alberto Ferreira

Published 2026-05-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luan Alberto Ferreira

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 walking along a very long, dark road made of numbers. Scattered along this road are special stones called prime numbers (like 2, 3, 5, 7, 11, etc.). For a long time, mathematicians have been trying to answer a simple question: If you take a small step forward on this road, will you definitely find a new prime stone?

This paper, written by Luan Alberto Ferreira, claims to have found a definitive "yes" for a specific type of step, solving a puzzle that has stumped mathematicians for decades.

Here is the breakdown of the paper's journey, using simple analogies:

1. The Goal: Finding Primes in Short Intervals

The author wants to prove that if you stand at a very large number $x$ and look ahead a distance of $x^\lambda$ (where $\lambda$ is a number between 0 and 1), you will find a predictable number of primes.

The Analogy: Imagine the road is so long that the stones (primes) get further apart as you go. The question is: If you take a "short" walk (a distance that is a fraction of your current position), do you still hit a stone?
The Claim: The paper proves that for any "short" step size defined by $x^\lambda$ , the number of primes you find is roughly equal to the length of your step divided by the natural logarithm of where you started. This confirms that primes are distributed evenly enough that you never go too long without finding one.

2. The Problem: The "Goldilocks" Weight

To prove this, the author uses a method inspired by a famous mathematician named Newman. Newman's method is like a metal detector for primes. To make the metal detector work, you need to attach a specific "weight" to every prime stone.

The Failed Attempts:
- Too Light: The author first tried using simple weights (like just the number itself). This was like using a weak magnet; it couldn't detect the stones in the short intervals because the signal was too faint.
- Too Heavy: Then, the author tried using exponential weights (like $e^p$ ). This was like using a nuclear-powered magnet; it was so strong it overwhelmed the detector, making it impossible to distinguish the specific pattern of the short intervals.
The Solution: The author found the "Goldilocks" weight. It is a special formula that grows faster than a polynomial but slower than a full exponential. It's a "just right" weight that amplifies the signal of primes in short intervals without drowning out the details.

3. The Method: The "Newman" Algorithm

The paper adapts a 10-step algorithm originally created by Newman to prove the Prime Number Theorem (which describes the general distribution of primes). The author tweaks this algorithm to work for these specific "short intervals."

The Magic Trick (The Weight Function): The author defines a special function $w(x)$ that assigns a value to each prime. By summing these values up to a certain point, they create a "cumulative score" ( $W(x)$ ).
The Proof: The paper shows that this cumulative score behaves exactly like a smooth, predictable curve ( $e^{cx^{1-\lambda}}$ ). Because the score matches the curve so perfectly, it forces the conclusion that the actual count of primes must also match the expected density.

4. The Technical Hurdle: The "Singularity" Wall

The hardest part of the proof (Steps 8–10) involves complex math called analytic continuation.

The Analogy: Imagine trying to walk across a bridge that has a hole in the middle (a mathematical "singularity" or a point where the math breaks down).
The Fix: The author initially tried to jump over the hole using a "second derivative" (a second step), but the bridge was still too shaky. The breakthrough came when the author decided to take a third step (a third derivative). This extra step acted like a stabilizer, smoothing out the rough edges of the math and allowing the proof to cross the gap safely. This allowed the author to prove the result without needing to assume the famous "Riemann Hypothesis" (a massive, unsolved problem in math).

5. The Results: Old Conjectures Solved

Because the author proved that primes are guaranteed to appear in these short intervals, several old, famous guesses about primes are now proven to be true (at least for very large numbers):

Legendre's Conjecture: There is always at least one prime number between any two consecutive square numbers (e.g., between $n^2$ and $(n+1)^2$ ). The paper proves this is true for all sufficiently large $n$ .
Sierpiński's Conjecture: If you arrange numbers $1$ to $n^2$ in a grid, every row contains at least one prime. This is also proven true for large $n$ .
Other Conjectures: Similar results apply to Brocard's and Oppermann's conjectures.

6. The Origin Story

The author shares a personal note on how the idea was born. It started with a desire to improve upon an old proof by Erdős. The author tried various mathematical "weights" to detect primes, failing with simple ones and overly complex ones, until they stumbled upon the specific exponential-weighted formula that worked. The final piece of the puzzle (the third derivative trick) came after months of struggling with a double-sum calculation that refused to cooperate.

Summary

In short, this paper is a mathematical detective story. The author built a custom "metal detector" (a specific weight function) and used a refined version of an old algorithm to prove that prime numbers are never too far apart, even in very short intervals. This settles long-standing questions about the distribution of primes and confirms that famous patterns in numbers hold true for the vast majority of the number line.

Technical Summary: Real Exponential Sums Over Primes and Prime Gaps

Problem Statement
The paper addresses the long-standing problem of the distribution of prime numbers in short intervals. Specifically, it seeks to prove that for any real number $\lambda$ such that $0 < \lambda < 1$ , the number of primes in the interval $(x, x + x^\lambda]$ satisfies the asymptotic relation:
$\pi(x + x^\lambda) - \pi(x) \sim \frac{x^\lambda}{\log(x)}$
where $\pi(x)$ is the prime counting function. This result would resolve several classical conjectures regarding the existence of primes between consecutive squares (Legendre's conjecture) and other related problems for sufficiently large $x$ .

Methodology
The author adapts D. J. Newman's elegant 1980 proof of the Prime Number Theorem (PNT) to the context of short intervals. The core of the methodology involves the following steps:

Weighted Summation: Instead of the standard Chebyshev function $\theta(x) = \sum_{p \le x} \log(p)$ , the author introduces a specific weight function $w(x)$ designed to handle the interval length $x^\lambda$ . The weight is defined as:
$w(x) = c(1-\lambda)\log(x) \exp(cx^{1-\lambda}) x^{-\lambda}$
for a parameter $c > 0$ . The cumulative sum is $W(x) = \sum_{p \le x} w(p)$ .
Reduction to Analytic Continuation: Following Newman's strategy, the proof reduces the asymptotic behavior of $W(x)$ to the convergence of an integral and the analytic continuation of a related Dirichlet series (or Laplace transform). The goal is to show that $W(x) \sim \exp(cx^{1-\lambda})$ .
Newman's Analytic Theorem: The proof utilizes Newman's theorem, which states that if a bounded, locally integrable function $h(x)$ has a Laplace transform $L(s)$ that extends analytically to the imaginary axis ( $\Re(s) \ge 0$ ), then the integral of $h(x)$ converges.
Handling Singularities via Mellin Transforms: A significant technical hurdle arises in Step 10 of the adaptation. The author defines a function $\tau(s)$ related to the Laplace transform of the weighted sum. To prove the necessary analytic continuation of $\tau(s) - 1/s$ to $\Re(s) \ge 0$ , the author employs the Mellin transform.
- The author initially attempts to use a second antiderivative but finds that the resulting integrand fails to converge absolutely on the boundary $\Re(s)=0$ due to the growth of the Gamma function terms.
- Key Technical Innovation: The author resolves this by utilizing the third antiderivative of the function. This introduces a quadratic denominator in the integrand (specifically $(z-1)(z-2)$ ), which ensures absolute convergence of the integral even after differentiation. This allows for the deformation of the integration contour into the critical strip without encountering non-integrable singularities.
Use of Zero-Free Regions: The proof relies on known zero-free regions of the Riemann zeta function (specifically the Vinogradov-Korobov type regions) to ensure the analytic properties of the auxiliary functions used in the Mellin inversion.

Key Contributions and Results

Main Theorem (Theorem 2.19): The paper proves that for any $0 < \lambda < 1$ , $\pi(x + x^\lambda) - \pi(x) \sim x^\lambda / \log(x)$ . This is established unconditionally, without assuming the Riemann Hypothesis.
Corollaries to Classical Conjectures: As a direct consequence of the main theorem, the paper demonstrates that for all sufficiently large $n$ $n$ :
- Legendre's Conjecture: There are at least two primes between $n^2$ and $(n+1)^2$ .
- Sierpiński's Conjecture: In an $n \times n$ matrix of integers from $1$ to $n^2$ , every row contains at least one prime.
- The paper also notes that Andrica's, Brocard's, and Oppermann's conjectures follow for sufficiently large integers.
Algorithmic Framework: The author outlines a general "algorithm" for solving prime distribution problems in intervals of length $\ell(x)$ by constructing appropriate weight functions $w(p)$ and analyzing their cumulative sums via analytic continuation.

Significance and Claims
The paper claims to solve a "long-standing problem" regarding the existence of primes in short intervals for the full range $0 < \lambda < 1$ . The author emphasizes that the solution relies on a careful adaptation of Newman's proof, specifically overcoming the technical difficulty of analytic continuation at the boundary $\Re(s)=0$ through the use of a third-order regularization (third antiderivative) in the Mellin transform domain.

The author is modest regarding the scope of the result, noting that while the asymptotic formula holds for large $x$ , it does not necessarily resolve the density of normalized prime gaps for all $n$ (e.g., Erdős's question about the sequence $(p_{n+1}-p_n)/\log p_n$ ). Furthermore, the paper acknowledges limitations imposed by Maier's Theorem, suggesting that the method may not extend to intervals growing slower than any power $x^\lambda$ (e.g., $(\log x)^\mu$ ). The author explicitly states that the proof of the main theorem is conditional on the validity of the analytic continuation steps derived, which are presented as the culmination of the work.

The paper concludes by reflecting on the origin of the weight function choice, tracing it from Erdős's binomial coefficient methods to Newman's PNT proof, and finally to the specific exponential weight required to balance the growth rates in short intervals.