On a new approach to the Riemann hypothesis

Assuming the Riemann hypothesis is false, this paper establishes an asymptotic relation between the residues of a specific series at nontrivial zeros and a continuous function of a rational parameter, offering a new approach to investigating the hypothesis.

The Gist

Imagine the Riemann Hypothesis as a massive, invisible fence running down the middle of a vast, foggy landscape. Mathematicians have long believed that all the "special points" (called zeros) of a very complex mathematical function called the Riemann Zeta-function are hiding exactly on this fence. If they are all on the fence, the universe of prime numbers behaves in a perfectly predictable, orderly way.

Hisanobu Shinya's paper asks a bold question: "What if the fence is wrong? What if some of these special points have wandered off into the fog?"

Instead of trying to prove the fence is perfect, Shinya decides to play a "What If" game. He assumes the hypothesis is false and sees what kind of chaos (or interesting patterns) would emerge from that mistake.

Here is the story of his approach, broken down into simple concepts:

1. The Detective's Tool: The "Prime Number Radar"

To investigate, Shinya builds a special mathematical tool called $M(s, p)$.

• The Analogy: Imagine you are trying to find hidden treasure (the zeros) in a dark forest. You have a radar that beeps when it detects "prime numbers" (the building blocks of all numbers).
• The Twist: Usually, you just scan the forest straight ahead. But Shinya's radar has a special dial called $p$ (a fraction like 1/2 or 3/7). By turning this dial, he changes the "frequency" of his scan. He is essentially asking: "If I look at the prime numbers through this specific colored lens, what happens?"

2. The "What If" Scenario: The Rogue Zero

Shinya assumes there is a "rogue" zero (a point where the function equals zero) that has strayed off the fence. Let's call this rogue $\rho^*$.

• The Assumption: This rogue is hiding in the fog, slightly to the right of the fence.
• The Goal: He wants to see if the behavior of his radar (the $M(s, p)$ function) changes in a way that reveals this rogue's presence.

3. The Great Equation: Connecting the Dots

The core of the paper is a massive, complex equation (Theorem 1.2 and 1.3).

• The Metaphor: Think of this equation as a bridge. On one side of the bridge is the "Rogue Zero" (the bad guy we assumed exists). On the other side is a collection of other mathematical objects (related to Dirichlet L-functions, which are like cousins of the Zeta-function).
• The Magic: Shinya proves that if the Rogue Zero exists, the two sides of the bridge must be perfectly balanced in a very specific, rhythmic way as the numbers get huge.

4. The "Smoothness" Surprise

Here is the most interesting part of the discovery.
Shinya finds that the formula describing this balance depends on the dial setting $p$.

• The Analogy: Imagine you are tuning a radio. As you slowly turn the dial from station A to station B, the music should change smoothly.
• The Problem: Shinya's formula suggests that the "music" (the mathematical relationship) should change smoothly as he turns the dial $p$.
• The Conflict: However, the math also suggests that if a Rogue Zero exists, the signal might get "jagged" or "broken" when the dial is set to certain fractions (rational numbers).

5. The Conclusion: A New Clue

The paper doesn't prove the Riemann Hypothesis is true or false. Instead, it sets a trap.

• The Takeaway: Shinya says, "If the Riemann Hypothesis is false, then this specific mathematical relationship must hold true, and it must be smooth as we change the dial $p$."
• The Catch: He admits that proving this relationship is actually smooth is very hard. It requires solving a new, difficult puzzle about how to count the "noise" in the radar signal when looking at huge numbers.

Summary for the Layperson

This paper is like a detective saying:

"I'm going to assume the suspect (the Riemann Hypothesis) is innocent, even though I think they are guilty. If they are innocent, then this specific fingerprint (the asymptotic relation) must appear on the wall. If we can prove that fingerprint doesn't exist, or if the wall is too messy to find it, then we might finally catch the suspect."

It's a reductio ad absurdum strategy: by assuming the opposite of what we believe, Shinya has derived a new, complex rule that must be true if we are wrong. Now, mathematicians have a new, strange rule to test. If they can prove this rule fails, they prove the Riemann Hypothesis is true. If they can prove the rule works, they might have found a crack in the foundation of number theory.

In short: It's a high-stakes mathematical game of "If I'm wrong, then this must happen," and the paper provides the blueprint for what "this" looks like.

Technical Summary

Here is a detailed technical summary of the paper "On a new approach to the Riemann hypothesis" by Hisanobu Shinya.

1. Problem Statement

The paper operates under the negation of the Riemann Hypothesis (RH). It assumes that the RH is false, implying the existence of non-trivial zeros of the Riemann $\zeta$-function (and associated Dirichlet $L$-functions) that lie off the critical line $\text{Re}(s) = 1/2$.

Specifically, the author investigates the consequences of assuming the existence of a zero $\rho^* = 1/2 + \eta^* + i\gamma^*$ with $\eta^* > 0$. The goal is to derive an asymptotic relationship involving the residues of a specific series at these off-critical zeros and to analyze the continuity of this relationship with respect to a rational parameter $p$.

2. Methodology

The author employs analytic number theory techniques, specifically focusing on Mellin transforms, residue calculus, and the properties of Dirichlet $L$-functions.

• The Core Function: The central object of study is the series:
  $$M(s, p) \equiv \sum_{n \ge 1} \Lambda(n) e^{-2\pi i p n} n^{-s}$$
  where $\Lambda(n)$ is the Mangoldt function and $p = a/b \in \mathbb{Q} \cap (0, 1)$.
  Using Lemma 1.1, the author establishes that $M(s, p)$ can be expressed as a linear combination of logarithmic derivatives of Dirichlet $L$-functions:
  $$M(s, a/b) = -\sum_{\chi} A(a, b; \chi) \frac{L'}{L}(s, \chi)$$
  where $A(a, b; \chi)$ is a coefficient involving Dirichlet characters.

• Integral Representation (Theorem 1.2): The author constructs a complex integral involving $M(s, p)$, Gamma functions, and exponential terms. By shifting the contour of integration and applying the residue theorem, the integral is evaluated in two ways:

  1. Summing residues at the poles of $M(s, p)$ (which correspond to the zeros of $L$-functions).
  2. Evaluating the integral via the logarithmic derivative of the Riemann $\zeta$-function ($-\zeta'/\zeta$) and auxiliary functions $K(p, \xi)$.

• Asymptotic Analysis: By equating the residue sum (LHS) with the integral evaluation (RHS) and taking the limit as the imaginary part of the zero ($\gamma_{n1,p}$) tends to infinity, the author derives an asymptotic equivalence.

3. Key Contributions and Results

Theorem 1.2 (Integral Identity)

The paper establishes a rigorous integral identity (Equation 2) relating a weighted sum of the Mangoldt function to a contour integral involving $-\zeta'/\zeta$ and Gamma functions. This serves as the foundational tool for the subsequent asymptotic analysis.

Theorem 1.3 (Main Result: Asymptotic Relation)

Assuming the existence of a pole $\rho_{n1,p} = 1/2 + \eta_{n1,p} + i\gamma_{n1,p}$ for $M(s, p)$ (where $\eta_{n1,p} > 0$ is the maximal real part among such poles in a specific region), the author derives the following asymptotic relation as $\gamma_{n1,p} \to \infty$:

$$c(p)\Gamma(\rho_{n1,p})e^{\pi\gamma_{n1,p}/2} \sim \Gamma(1 + \nu + i\gamma_{n1,p})e^{\pi\gamma_{n1,p}/2} \sum_{\rho_j} \rho_j^{-1} (2\pi p)^{-\nu - i\gamma_{n1,p} + \rho_j} e^{(\nu + i\gamma_{n1,p} - \rho_j)\pi i/2} \Gamma(-\nu - i\gamma_{n1,p} + \rho_j)$$

Key properties of this result:

• Continuity: The expression on the right-hand side is continuous with respect to the rational parameter $p \in (0, 1)$.
• Implication: Since the left-hand side depends on the residue $c_{\rho_{n1,p}}$ and the pole location, the continuity of the RHS suggests that the behavior of the residues and poles must vary continuously with $p$.

Conjecture 1.4 (Bounding Problem)

The author identifies a critical gap in the proof required to fully validate the asymptoticity: bounding the sum of terms associated with $L$-function zeros for large denominators $b$.
The conjecture states that for arbitrary $\epsilon > 0$:
$$\sum_{\chi} A(a, b; \chi) \sum_{|t-\gamma|<1} \frac{1}{s - \rho_\chi} \ll t^\epsilon$$
as $t \to \infty$, with the constant independent of $b$. Resolving this is presented as necessary to fully understand the implications of the derived relation for the Riemann Hypothesis.

4. Significance and Implications

• New Perspective on RH Negation: Rather than simply searching for counterexamples, this paper constructs a specific functional relationship that must hold if the RH is false. It links the distribution of off-critical zeros to the arithmetic properties of rational numbers $p$.
• Continuity Argument: The derivation suggests that if off-critical zeros exist, their residues and positions must exhibit a specific continuity with respect to the rational parameter $p$. This provides a new potential avenue for contradiction: if one can show that such continuity is impossible or leads to a divergence, it would imply the RH is true.
• Methodological Innovation: The paper introduces the series $M(s, p)$ (a twisted sum of the Mangoldt function) as a tool for studying the Weak Goldbach Conjecture and the RH, showing how it can be decomposed into Dirichlet $L$-functions.
• Open Problems: The paper explicitly highlights the difficulty in bounding the sum over characters for large $b$ (Conjecture 1.4). Solving this bound is the key to turning the derived asymptotic relation into a definitive proof or disproof of the RH.

Summary

Hisanobu Shinya's paper proposes a novel analytical framework assuming the falsity of the Riemann Hypothesis. By analyzing the series $M(s, p)$, the author derives an asymptotic formula connecting the residues of off-critical zeros to a continuous function of a rational parameter. The work suggests that the existence of such zeros would impose strict continuity constraints on the distribution of $L$-function zeros, offering a new reductio ad absurdum approach, contingent upon resolving specific bounding estimates for character sums.