🔢 mathematics

The exceptional set of Goldbach problem and Linnik's constant

This paper establishes that the number of even integers up to $X$ that cannot be expressed as the sum of two primes is bounded by $O(X^{7/10})$ and simultaneously proves that the least prime in an arithmetic progression modulo $q$ is bounded by $O(q^5)$ , utilizing a unified method for both results.

Original authors: Genheng Zhao

Published 2026-01-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Genheng Zhao

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 Missing Puzzle Piece

Imagine a giant puzzle called the Goldbach Conjecture. The rule of the puzzle is simple: "Take any even number bigger than 2 (like 4, 6, 100, 1,000,000), and you should be able to build it by adding exactly two prime numbers together." (Primes are numbers like 2, 3, 5, 7, 11 that can't be divided by anything else).

Mathematicians have been trying to solve this puzzle for centuries. They are pretty sure it's true, but they can't prove it for every single number yet.

So, instead of trying to prove it for everyone, they ask: "How many people are breaking the rules?"
In this paper, the author is counting the "rule-breakers." These are the even numbers that cannot be made by adding two primes. Let's call this group the Exceptional Set (the "bad apples").

The goal of this paper is to prove that this "bad apple" group is actually quite small. Specifically, the author proves that if you look at all even numbers up to a huge number $X$ , the number of "bad apples" is roughly less than $X^{0.7}$ . (For comparison, if the rule-breakers were as common as the numbers themselves, the count would be $X^1$ . If they were as rare as squares, it would be $X^{0.5}$ . This result puts the bad apples somewhere in between, but much closer to the "rare" side).

The Tool: The "Prime Detective" (L-functions)

How do you count these missing numbers? You can't just check them one by one; there are too many. Instead, mathematicians use a sophisticated tool called L-functions.

Think of an L-function as a super-sensitive metal detector for prime numbers.

When the metal detector is working perfectly, it beeps loudly for every prime.
However, sometimes the detector gets "glitchy." It might miss a prime or give a false signal. In math terms, these glitches are called Zeroes.

The author's job is to analyze these glitches. If there are too many glitches in a specific area, it means there are many "bad apples" (exceptions to the Goldbach rule). If the glitches are sparse, the "bad apples" are rare.

The Strategy: Catching the Glitches

The paper is essentially a high-stakes game of "Whac-A-Mole" with these mathematical glitches. The author uses a refined version of a method developed by a mathematician named Pintz.

Here is the analogy:
Imagine the glitches (Zeroes) are moles popping up in a garden.

The Old Method: Previous mathematicians could prove that the moles were somewhat rare, but they had to leave a little bit of space in the garden where moles could hide.
The New Method: Zhao refines the "whack" (the mathematical calculation). He tightens the rules on where the moles can hide. He proves that even if a mole pops up, it can't be too close to the center of the garden without causing a contradiction.

By squeezing the space where these "glitches" are allowed to exist, the author forces the number of "bad apples" (exceptions to Goldbach) to shrink significantly.

The Second Achievement: The "Fastest Prime" (Linnik's Constant)

The paper also tackles a related problem called Linnik's Constant.

Imagine you have a row of mailboxes numbered 1, 2, 3... up to $q$ . You want to find the first mailbox that contains a "Prime Letter."

The Problem: How far down the row do you have to go before you are guaranteed to find a prime?
The Constant ( $L$ ): Mathematicians have a number $L$ that says, "You will definitely find a prime by mailbox number $q^L$ ."

For a long time, the best known value for $L$ was 5.2. This meant you might have to check up to $q^{5.2}$ mailboxes.

The Breakthrough: Using the same refined "metal detector" logic, the author shows that you only need to check up to $q^5$ .
The Metaphor: It's like shrinking the search area for a lost key. Previously, you had to search a whole city block ( $q^{5.2}$ ). Now, the author proves you only need to search a single city square ( $q^5$ ). It's a small numerical change, but in the world of prime numbers, it's a massive leap forward.

Summary of the Results

Goldbach Exceptions: The author proves that the number of even numbers that fail the Goldbach rule is very small (specifically, bounded by $X^{0.7}$ ). This is a tighter, better bound than what was known before.
Linnik's Constant: The author improves the guarantee for finding the first prime in a sequence, lowering the exponent from 5.2 to 5.

Why This Matters (Without the Hype)

The paper doesn't claim to solve the Goldbach Conjecture (we still don't know if every even number works). It doesn't claim to help with cryptography or computer security directly.

Instead, it is a pure mathematics victory. It shows that our "metal detectors" for prime numbers are getting sharper. By squeezing the mathematical rules tighter, the author has reduced the "wiggle room" for exceptions and improved our understanding of how primes are distributed. It's like a surveyor refining a map: the territory hasn't changed, but our measurement of it is now more precise.

Technical Summary: The Exceptional Set of Goldbach Problem and Linnik's Constant

Problem Statement
The paper addresses two fundamental problems in analytic number theory: the size of the exceptional set in the Goldbach conjecture and the value of Linnik's constant.

Goldbach's Exceptional Set: Let $E(X)$ denote the number of even integers $m \le X$ that cannot be expressed as the sum of two odd primes. The goal is to establish a strong upper bound for $E(X)$ .
Linnik's Constant: Let $P(q)$ denote the least prime in an arithmetic progression $\{a, a+q, a+2q, \dots\}$ where $(a,q)=1$ . The goal is to determine the smallest admissible constant $L$ such that $P(q) = O(q^L)$ .

Historically, bounds for $E(X)$ have evolved from conditional results under the Generalized Riemann Hypothesis to unconditional bounds involving the Siegel zero. The breakthrough by Montgomery and Vaughan established $E(X) = O(X^{1-\delta})$ for some $\delta > 0$ , a result deeply connected to Linnik's method for bounding $P(q) = O(q^L)$ . Prior to this work, the best known admissible values were $\delta = 0.28$ (Pintz) and $L = 5.2$ (Xylouris).

Methodology
The paper employs a refinement of Pintz's method, which relies on the distribution of zeros of Dirichlet $L$ -functions near the line $\sigma = 1$ . The core of the argument involves analyzing the zeros $\varrho = \beta + i\gamma$ in the critical region $\sigma \in [1 - H(\log q)^{-1}, 1]$ and $|t| \le H$ .

The methodology proceeds through the following technical steps:

Reduction to Zero Sums: Following Pintz's reduction, the problem is reduced to estimating sums over zeros of $L$ -functions for a single modulus $q$ . The paper defines a multi-set of zeros $Z(\chi, H)$ and seeks to bound the sum of squared weighted sums of these zeros.
Weighted Sums and Laplace Transforms: The author utilizes specific test functions derived from Heath-Brown's work, specifically involving the function $f_1(t)$ and its Laplace transform $F_1(z)$ , and a related function $F_2(z)$ . These are used to construct inequalities involving the real parts of sums over zeros.
Zero Density Estimates: The paper employs a generalized zero density estimate (attributed to Pintz's generalization of Heath-Brown's work) which counts all zeros in a specific region rather than just those belonging to $L$ -functions with at least one zero in that region. This avoids the use of suboptimal functions found in earlier methods.
Case Analysis: The proof of the main theorem is divided into several cases based on the magnitude of the smallest real part of the zeros, denoted $\lambda_{1,1} = \min \text{Re}(1-\varrho)$ . The author uses Lemma 2.4 to establish bounds on the number of zeros $N(\Lambda)$ in specific regions depending on the value of $\lambda_{1,1}$ .
Optimization of Parameters: The author introduces parameters $x, y, z, \Lambda, \lambda_0$ to optimize the bounds on weighted sums of the form $\sum e^{-k \max\{\lambda_j, \Lambda\}}$ . By carefully selecting these parameters and utilizing the Deuring-Heilbronn phenomenon (which limits the number of exceptional zeros), the author derives tighter inequalities for the sums of exponentials of the zeros.

Key Contributions and Results
The primary contribution of the paper is the improvement of the admissible constants for both the Goldbach exceptional set and Linnik's constant through a refined analysis of Pintz's method.

Theorem 1.1 (Goldbach Exceptional Set): The paper proves that $E(X) = O(X^{7/10})$ . This corresponds to an admissible value of $\delta = 0.3$ in the bound $E(X) = O(X^{1-\delta})$ . This improves upon the previous best result of $\delta = 0.28$ .
- The proof relies on establishing Theorem 1.2, which shows that for a set of characters satisfying certain conditions, the sum of squared weighted sums of zeros is strictly less than 1 (specifically $\le 1 - c_1$ ).
- The improvement is achieved by reducing the exponent in the key inequality from $25/7$ (used by Pintz) to $10/3$ . This reduction is attributed to a refined estimate of sums of the type $\sum e^{-10/3 \max\{\lambda, \Lambda\}}$ and a more precise estimation of the difference terms.
Theorem 1.4 (Linnik's Constant): The paper demonstrates that $L = 5$ is an admissible constant for Linnik's problem, improving upon the previous record of $L = 5.2$ .
- This result is derived from Theorem 1.3, which establishes a bound on the sum of exponentials of zeros ( $\sum e^{-5\lambda} \le 1 - c_1$ ).
- The improvement from $L=5.2$ to $L=5$ is explicitly attributed to the use of Pintz's generalized zero density estimate, which counts all zeros in the region rather than just the count of $L$ -functions with zeros there. This allows for the avoidance of less optimal functions used in Heath-Brown's original method.

Significance
The paper claims that its significance lies in the "refinement of Pintz's method." By optimizing the parameters and the specific functions used to estimate sums over zeros, the author uncovers the potential of Pintz's framework to push the boundaries of these constants further. The improvement of $\delta$ from $0.28$ to $0.3$ and $L$ from $5.2$ to $5$ represents a quantitative tightening of the bounds on the distribution of primes in arithmetic progressions and the exceptions to the Goldbach conjecture. The author notes that the implicit constants in these bounds are ineffective, a standard limitation in methods relying on the exclusion of Siegel zeros.