Minimal zero-free regions for results on primes between consecutive perfect $k$th powers

This paper computes minimal zero-free regions for the Riemann zeta-function to prove the existence of a prime between consecutive perfect $k$th powers for $k \geq 65$, specifically establishing this result for $k=86$ and a specific subset of integers for $k=70$, while quantifying the progress toward Legendre's conjecture.

The Gist

Imagine you are a detective trying to solve a mystery about prime numbers (numbers like 2, 3, 5, 7, 11 that can only be divided by 1 and themselves).

For over a century, mathematicians have been obsessed with a specific question: Is there always at least one prime number hiding between two consecutive perfect squares?

Think of perfect squares as stepping stones: $1^2, 2^2, 3^2, 4^2$ (which are 1, 4, 9, 16). The mystery is: Is there always a prime number in the gap between stone 4 and stone 9? Between stone 9 and stone 16? And so on, forever?

This is called Legendre's Conjecture. It's a famous "unsolved case." We are pretty sure the answer is "Yes," but nobody has been able to write a proof that works for every single number in existence.

The Author's New Strategy: "Stretching the Gap"

Since proving it for squares (power of 2) is too hard right now, the author, Ethan Simpson Lee, decided to try a different approach. He asked: "What if we make the gaps between our stepping stones much bigger?"

Instead of looking at $n^2$ and $(n+1)^2$, let's look at $n^{86}$ and $(n+1)^{86}$.

• The Analogy: Imagine the squares are tiny pebbles. The 86th powers are massive boulders. The gap between two boulders is huge. It's much easier to find a prime number hiding in a giant canyon than in a tiny crack between pebbles.

The author's goal was to find the smallest possible gap (the lowest power $k$) where we can guarantee a prime exists, without needing to check every single number by hand.

The Three Tools of the Trade

To solve this, the author used a high-tech toolkit involving the Riemann Zeta-function (a complex mathematical machine that acts like a "prime number detector"). To prove his results, he needed three specific ingredients:

1. Zero-Free Zones: Imagine the Zeta-function has "ghosts" (zeros) that float around. If a ghost lands in a specific dangerous zone, it messes up our prime count. The author had to prove these ghosts stay out of a specific "safe zone."
2. Ghost Counting: He needed to count exactly how many ghosts could possibly be in a certain area.
3. Error Margins: Since he's using approximations, he had to calculate exactly how much his math could be "off" and prove that even with the worst-case error, a prime still exists.

The Big Discoveries

Using these tools and some very powerful computers, the author achieved three main things:

1. The "86th Power" Victory
He proved that for any number $n$, there is definitely a prime number between $n^{86}$ and $(n+1)^{86}$.

• Why it matters: Before this, the best known result was for the 90th power. He tightened the gap from 90 down to 86. It's like saying, "We can guarantee a treasure chest exists in any canyon wider than 86 miles," whereas before we could only guarantee it for 90-mile canyons.

2. The "70th Power" Loophole
He couldn't prove it for every number with the 70th power yet. However, he found a clever trick. He proved that if you skip the very first few numbers (specifically, if you start looking at numbers larger than a specific, astronomically huge number), then yes, there is always a prime between the 70th powers.

• The Metaphor: It's like saying, "We can't guarantee a prime in the first few miles of the road, but once you pass mile 98, you are guaranteed to find one every 70 miles."

3. The "What-If" Map (The Zero-Free Regions)
This is the most exciting part for future detectives. The author created a map showing exactly how "ghost-free" the Zeta-function needs to be to prove the 70th power (or 75th, 80th, etc.) works for every number.

• The Analogy: He didn't just say "It's hard." He said, "If we can prove the ghosts stay out of this specific thin rectangle on the map, then the 70th power mystery is solved."
• This tells other mathematicians exactly where to focus their energy. It's like a treasure map saying, "Dig here, and you'll find the proof."

Why Should You Care?

You might think, "Who cares about 86th powers?"

• The Milestone: Legendre's Conjecture (the square gap) is the "Holy Grail." Every time we prove it for a higher power (like 86, then 85, then 80...), we are inching closer to the Holy Grail.
• The Progress: This paper shows us exactly how far we are from solving the ultimate mystery. It quantifies the difficulty. It tells us, "We are very close, but we need to push the 'ghost-free' zone just a tiny bit further."

Summary

Ethan Simpson Lee is a detective who used a powerful mathematical machine to prove that prime numbers are guaranteed to exist in very large gaps between massive numbers. He lowered the "guarantee threshold" from 90 to 86, found a way to guarantee it for 70 (if we ignore the very beginning), and drew a map for other detectives to follow if they want to solve the case for even smaller gaps.

It's a step-by-step march toward solving one of the oldest and most famous riddles in mathematics.

Technical Summary

Here is a detailed technical summary of the paper "Minimal Zero-Free Regions for Results on Primes Between Consecutive Perfect $k$th Powers" by Ethan Simpson Lee.

1. Problem Statement

The paper addresses a fundamental problem in analytic number theory: determining the smallest integer $k \ge 3$ such that there is guaranteed to be at least one prime number between consecutive perfect $k$-th powers, i.e., in the interval $(n^k, (n+1)^k)$ for all integers $n \ge 1$.

• Context: Legendre's Conjecture posits that there is always a prime between consecutive squares ($k=2$). This remains unproven even under the assumption of the Riemann Hypothesis (RH).
• Current State: Ingham (1937) proved the existence of a prime for $k=3$ for sufficiently large $n$. Recent work by Cully-Hugill established this for $n \ge \exp(\exp(32.892))$ with $k=3$. The best unconditional result for all $n \ge 1$ (without assuming RH) previously stood at $k=90$.
• Goal: The author aims to lower the admissible value of $k$ for which the result holds for all $n \ge 1$, and to quantify the "distance" to proving this for smaller $k$ (specifically $k < 86$) by computing the minimal zero-free regions required for the Riemann zeta-function $\zeta(s)$.

2. Methodology

The proof relies on three main components:

1. Zero-Free Regions: Establishing regions in the complex plane where $\zeta(\sigma + it) \neq 0$. The paper utilizes the "Littlewood form" of zero-free regions:
   $$\sigma \ge 1 - \frac{\log \log |t|}{Z \log |t|}$$
2. Zero-Density Estimates: Using upper bounds for $N(\sigma, T)$, the number of non-trivial zeros with real part $\beta > \sigma$ and imaginary part $0 < \gamma < T$. The author combines explicit bounds from Bellotti and others.
3. Explicit Error Bounds: Refining the error terms in the truncated Riemann–von Mangoldt formula and the difference of Chebyshev functions $\theta(x)$.

Key Technical Innovation:
The author introduces a novel strategy to bridge the gap between "sufficiently large $n$" and "all $n \ge 1$":

• Proposition 3.1: A new, parametrized bound for the difference $\theta(x+h) - \theta(x) - h$. This bound is designed to be flexible, allowing the author to switch between different zero-density estimates depending on the range of $x$ and the specific $k$ being analyzed.
• Hybrid Approach: The proof splits the range of $x$ (where $x = n^k$) into three segments:
  1. Small $x$: Verified computationally or via existing explicit prime gap results (Proposition 3.3).
  2. Intermediate $x$: Proven using standard zero-free regions and density estimates (Proposition 3.2).
  3. Large $x$ (The Gap): For $k < 86$, there is a gap between the intermediate and large ranges. The author proves that if the zero-free region is sufficiently "thin" (i.e., if specific constants $Z_k$ and $T_k$ are met), the result holds.
• Subset Strategy (Theorem 1.3): For $k=70$, instead of proving the result for all $n$, the author constructs a specific subset of integers $a_n = n(1 + \lfloor N/n \rfloor)$ where the result holds unconditionally, provided $N$ is sufficiently large.

3. Key Contributions and Results

Theorem 1.2: Improvement to $k=86$

The author proves that for every integer $n \ge 1$, there is at least one prime in the interval $(n^{86}, (n+1)^{86})$.

• This improves the previous best unconditional bound of $k=90$.
• The proof combines Proposition 3.2 (for large $x$) and Proposition 3.3 (for intermediate $x$) to cover the entire range $x \ge 3.68 \times 10^{19}$.

Theorem 1.3: A Subset Result for $k=70$

The author proves that there is a prime between consecutive 70th powers for a specific sequence of integers $a_n$, provided:
$$N \ge \exp\left(\frac{15951}{70}\right) - \exp\left(\frac{2135}{70}\right) \approx 9.189 \times 10^{98}$$

• This demonstrates that the result for $k=70$ is "almost" true for all $n$, failing only for a finite (though astronomically large) initial segment of integers.

Theorem 1.4: Minimal Zero-Free Regions

This is the paper's most significant theoretical contribution. The author computes the minimal zero-free regions required to prove the result for $k \in \{85, 80, 75, 70\}$ for all $n \ge 1$.

• Condition: If $\zeta(\sigma + it) \neq 0$ in the region:
  $$1 - \frac{\log \log |t|}{Z_k \log |t|} \le \sigma \le 1 \quad \text{and} \quad H_R < |t| \le T_k$$
  then a prime exists between $n^k$ and $(n+1)^k$ for all $n \ge 1$.
• Specific Constants (Table 1):
  • $k=85$: Requires $Z_{85} = 17.270$ and $T_{85} = e^{75.853}$.
  • $k=70$: Requires $Z_{70} = 14.055$ and $T_{70} = e^{264.334}$.
• Significance: This quantifies the difficulty of proving Legendre's conjecture milestones. For example, proving the result for $k=70$ would require verifying the zero-free region up to a height of $|t| \approx e^{264}$, which is computationally challenging but theoretically conceivable.

4. Significance and Implications

1. Quantifying the Gap: The paper moves beyond simply stating "we need better bounds" to providing explicit, minimal requirements for the zero-free regions of the Riemann zeta-function. It tells researchers exactly how much the zero-free region needs to be improved to solve the problem for specific $k$.
2. Methodological Flexibility: The parametrized bounds in Proposition 3.1 allow for the integration of future improvements in zero-density estimates ($N(\sigma, T)$) without re-deriving the entire proof structure.
3. Feasibility Analysis: By calculating that $k=70$ requires a zero-free region up to $|t| \approx e^{264}$, the paper provides a concrete target for computational number theorists. It highlights that while $k=70$ is currently out of reach, the gap is not infinite; it is a matter of computational power and refined constants.
4. Subset Results: The strategy of proving results for a "dense" subset of integers (Theorem 1.3) offers a new pathway for tackling similar problems where a complete proof for all $n$ is currently impossible.

5. Conclusion

Ethan Simpson Lee's paper represents a significant advancement in the explicit study of prime gaps between perfect powers. By lowering the unconditional bound to $k=86$ and mapping the precise zero-free regions required for $k=70$, the paper bridges the gap between theoretical conjectures and explicit computational verification, providing a clear roadmap for future progress toward Legendre's Conjecture.