A new equivalence to the Riemann Hypothesis by means of the Salem integral equation

Based on the provided paper, here is a detailed technical summary of the work by B. J. González and E. R. Negrín.

1. Problem Statement

The paper addresses the Riemann Hypothesis (RH), one of the most famous unsolved problems in mathematics. The hypothesis asserts that all non-trivial zeros of the Riemann zeta-function $\xi(s)$ (specifically within the critical strip $0 < \text{Re}(s) < 1$) lie on the critical line $\text{Re}(s) = \frac{1}{2}$.

The authors aim to establish a new equivalence to the Riemann Hypothesis. They build upon a 1953 result by Raphaël Salem, which states that the RH is true if and only if the only bounded measurable function $f(t)$ satisfying a specific integral equation for all $x > 0$ is the trivial solution $f = 0$ almost everywhere. The authors seek to refine this condition by restricting the search for non-trivial solutions to a specific class of functions.

2. Methodology

The authors utilize the relationship between the Riemann zeta-function ($\xi(s)$), the Dirichlet eta-function ($\eta(s)$), and a specific Salem integral equation.

• Definitions:
  • $\xi(s) = \sum_{n=1}^{\infty} n^{-s}$ for $\text{Re}(s) > 1$.
  • $\eta(s) = \sum_{n=1}^{\infty} (-1)^{n-1} n^{-s}$ for $\text{Re}(s) > 0$.
  • The relation $\eta(s) = (1 - 2^{1-s})\xi(s)$ connects the two functions, allowing the extension of $\xi(s)$ to the critical strip.
• Integral Representation:
  The authors rely on the known integral identity valid for $\text{Re}(s) > 0$ and $x > 0$:
  $$\int_0^{\infty} \frac{t^{s-1}}{e^{xt} + 1} \, dt = x^{-s}\Gamma(s)\eta(s)$$
• Logical Derivation:
  1. They substitute $s = \delta + i\gamma$ (where $0 < \delta < 1$ and $\gamma \in \mathbb{R}$) into the integral identity.
  2. They observe that $s$ is a zero of $\xi(s)$ if and only if $\eta(s) = 0$ (since the factor $1-2^{1-s}$ is non-zero for non-trivial zeros).
  3. Consequently, $s = \delta + i\gamma$ is a zero of $\xi(s)$ if and only if:
     $$\int_0^{\infty} \frac{t^{\delta-1} t^{i\gamma}}{e^{xt} + 1} \, dt = 0 \quad \text{for all } x > 0$$
  4. This integral equation corresponds to the general form $\int_0^{\infty} \frac{t^{\delta-1}}{e^{xt} + 1} f(t) \, dt = 0$ with the specific test function $f(t) = t^{i\gamma}$.

3. Key Contributions

The primary contribution of the paper is the formulation of a restricted equivalence to the Riemann Hypothesis.

• Refinement of Salem's Condition: While Salem's original condition requires that no bounded measurable function (other than zero) satisfies the integral equation, González and Negrín prove that it is sufficient to check only functions of the specific oscillatory type $f(t) = t^{i\gamma}$ (which are bounded measurable functions with modulus 1).
• The New Equivalence: The Riemann Hypothesis is true if and only if for every $\delta$ in the range $\frac{1}{2} < \delta < 1$, the integral equation
  $$\int_0^{\infty} \frac{t^{\delta-1}}{e^{xt} + 1} f(t) \, dt = 0 \quad (\forall x > 0)$$
  has no solutions of the form $f(t) = t^{i\gamma}$ for any real $\gamma$.

4. Results

The paper establishes the following logical chain:

1. If the Riemann Hypothesis is false, there exists a zero $s = \delta + i\gamma$ with $\delta \neq \frac{1}{2}$. Due to the symmetry of zeros about the line $\text{Re}(s) = \frac{1}{2}$, we can assume $\frac{1}{2} < \delta < 1$.
2. The existence of such a zero implies that the function $f(t) = t^{i\gamma}$ is a non-trivial solution to the Salem integral equation for that specific $\delta$.
3. Conversely, if the integral equation has a solution of the form $f(t) = t^{i\gamma}$ for some $\frac{1}{2} < \delta < 1$, then $\eta(\delta + i\gamma) = 0$, implying a zero off the critical line, thus falsifying the RH.
4. Conclusion: The RH holds if and only if the integral equation admits no power-law oscillatory solutions $t^{i\gamma}$ for $\delta \in (\frac{1}{2}, 1)$.

5. Significance

• Simplification of Verification: The result significantly narrows the scope of functions that must be analyzed to verify the Riemann Hypothesis via the Salem integral equation. Instead of investigating the entire space of bounded measurable functions, one only needs to investigate the specific subspace of functions $f(t) = t^{i\gamma}$.
• Theoretical Insight: It highlights a direct link between the spectral properties of the Salem integral operator and the location of the zeros of the zeta function. It suggests that the "non-existence" of RH-violating zeros is equivalent to the "non-existence" of specific oscillatory modes in the kernel of the integral operator.
• Connection to Existing Literature: The paper bridges the gap between the general functional-analytic approach of Salem (1953) and the specific arithmetic nature of the zeta zeros, providing a more concrete target for potential counter-examples or proofs.

Note: The paper is presented as a short note establishing this equivalence. It does not provide a proof of the Riemann Hypothesis itself, but rather a new necessary and sufficient condition for its truth.