Pathway Towards a Proof of the Riemann Hypothesis via Spectral Transfer

1. Introduction

Since its formulation by Bernhard Riemann in 1859, the Riemann Hypothesis (RH) has been a central pillar of analytic number theory [1]. It postulates that the zeros of the Zeta function, $\zeta \left(s\right)={\sum }_{n=1}^{\infty }{n}^{-s}$, in the critical strip $0<\mathfrak{R}\left(s\right)<1$, all lie on the line $\mathfrak{R}\left(s\right)=1/2$. Despite vast numerical evidence and deep connections to the distribution of prime numbers [2,3], a formal proof remains elusive.

The strategy of this work departs from traditional approaches and explores a path of "property transfer." It is a known fact that analogues of the RH have been proven for function fields over finite fields and also for certain Dirichlet L-functions and Dedekind Zeta functions. In particular, it has been proven that the Dedekind Zeta function of the field of Gaussian integers, ${\zeta }_{\mathbb{Q}\left(i\right)}\left(s\right)=\zeta \left(s\right)L(s,{\chi }_4)$, where ${\chi }_4$ is the non-trivial Dirichlet character modulo 4, satisfies the RH [5]. That is, ${\zeta }_{\mathbb{Q}\left(i\right)}\left(s\right)$ has no zeros in the region $\mathfrak{R}\left(s\right)>1/2$.

This paper proposes a mechanism to "transfer" this property from the field $\mathbb{Q}\left(i\right)$ to the field $\mathbb{Q}$. To do this, we construct a spectral transfer operator $V\left(s\right)$ which, by design, must be a holomorphic function if certain conditions are met. The hypothetical existence of a zero of $\zeta \left(s\right)$ off the critical line would induce a singularity in $V\left(s\right)$, leading to a contradiction.

2. Methodology: Construction of the Transfer Operator

The key to our proposal is the construction of an analytical bridge between the worlds where the RH is known and where it is a conjecture.

2.1. The Critical Functional Object

To relate both Zeta functions, we use their completed versions (Xi-functions), which are entire functions and satisfy a simple functional equation. The Riemann Xi-function is:

$${\xi }_{\mathbb{Q}}\left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\mathsf{\Gamma }\left({\displaystyle \frac{s}{2}}\right)\zeta \left(s\right)$$

Analogously, for the Gaussian integers, we have:

$${\xi }_{\mathbb{Q}\left(i\right)}\left(s\right)={\left(2\pi \right)}^{-s}\mathsf{\Gamma }\left(s\right){\zeta }_{\mathbb{Q}\left(i\right)}\left(s\right)$$

We define our functional transfer object as the quotient of these two functions, regularized by an exponential factor:

Definition 1 

(Spectral Transfer Operator). The operator $V\left(s\right)$ is defined as:

$$V\left(s\right)={\displaystyle \frac{{\xi }_{\mathbb{Q}\left(i\right)}\left(s\right)}{{\xi }_{\mathbb{Q}}\left(s\right)}}{e}^{-g\left(s\right)}$$

(1)

where $g\left(s\right)$ is a "corrector crutch" function designed to cancel spurious singularities or zeros that arise from the structural differences between $\mathbb{Q}$ and $\mathbb{Q}\left(i\right)$.

2.2. The Corrector Crutch $g\left(s\right)$

The relationship between both Zeta functions is given through their Euler products:

$${\zeta }_{\mathbb{Q}\left(i\right)}\left(s\right)=\prod _p{(1-p^{-s})}^{-1}\prod _p{(1-{\chi }_4\left(p\right)p^{-s})}^{-1}=\zeta \left(s\right)L(s,{\chi }_4)$$

The factor $L(s,{\chi }_4)$ introduces differences in poles and analytical structure. The function $g\left(s\right)$ must be holomorphic in the region of interest and precisely compensate for these differences so that $V\left(s\right)$ has the opportunity to be holomorphic. Its construction is non-trivial and is based on the contributions of primes $p\equiv 3(mod4)$. A possible form is:

$$g\left(s\right)=\sum _{p\equiv 3(mod4)}log\left({\displaystyle \frac{1-p^{-2s}}{1-p^{-s}}}\right)+\dots$$

(2)

The convergence and holomorphy of $g\left(s\right)$ for $\mathfrak{R}\left(s\right)>1/3$ is a fundamental technical pillar of this proposal.

2.3. Central Theorem of the Proposal

Our argument is consolidated in the following conditional theorem:

Theorem 1

(Transfer Theorem). If the function $g\left(s\right)$ is correctly defined to be holomorphic for $\mathfrak{R}\left(s\right)>1/2$ and if the resulting operator $V\left(s\right)$ defined in Equation (1) is holomorphic and bounded in the region $\mathfrak{R}\left(s\right)>1/2$, then the Riemann Hypothesis is true.

Proof 

(Sketch of Proof). Assume, for the sake of contradiction, that there exists a zero $\rho$ of $\zeta \left(s\right)$ such that $\mathfrak{R}\left(\rho \right)>1/2$. By the definition of the Xi-function, ${\xi }_{\mathbb{Q}}\left(\rho \right)=0$. However, since the RH is true for ${\zeta }_{\mathbb{Q}\left(i\right)}\left(s\right)$, we know that ${\xi }_{\mathbb{Q}\left(i\right)}\left(\rho \right)\ne 0$.

When evaluating $V\left(s\right)$ at $s=\rho$, the denominator ${\xi }_{\mathbb{Q}}\left(\rho \right)$ vanishes, while the numerator ${\xi }_{\mathbb{Q}\left(i\right)}\left(\rho \right)$ and the factor $e^{-g\left(\rho \right)}$ are finite and non-zero. This implies that $V\left(s\right)$ would have a simple pole at $s=\rho$.

The existence of such a pole contradicts the premise that $V\left(s\right)$ is holomorphic for $\mathfrak{R}\left(s\right)>1/2$. Therefore, our initial assumption must be false: no zero of $\zeta \left(s\right)$ can exist with $\mathfrak{R}\left(s\right)>1/2$. By the functional equation of $\zeta \left(s\right)$, this also excludes zeros in $0<\mathfrak{R}\left(s\right)<1/2$. □

3. Analysis of the Transfer Operator

The validity of this proposal rests on proving the holomorphy and boundedness of $V\left(s\right)$.

• Holomorphy: Since ${\xi }_{\mathbb{Q}\left(i\right)}\left(s\right)$ has no zeros for $\mathfrak{R}\left(s\right)>1/2$ and $g\left(s\right)$ is constructed to be holomorphic, the only potential source of singularities for $V\left(s\right)$ are the zeros of ${\xi }_{\mathbb{Q}}\left(s\right)$. The argument of the Transfer Theorem shows that this situation is precisely what is sought to be excluded.
• Boundedness: For the argument to be robust (e.g., avoiding removable singularities), it is crucial that $V\left(s\right)$ is bounded. Using the asymptotic estimates of the Gamma function (Stirling’s formula) and the known behavior of Xi-functions in vertical strips [6], it can be argued that for $|t|=\left|\mathfrak{I}\right(s\left)\right|\to \infty$:

  $$|{\xi }_{\mathbb{Q}}\left(s\right)|\sim C_1{\left|t\right|}^{A_1}{e}^{-{\displaystyle \frac{\pi }{4}}\left|t\right|}\mathrm{and}|{\xi }_{\mathbb{Q}\left(i\right)}\left(s\right)|\sim C_2{\left|t\right|}^{A_2}{e}^{-{\displaystyle \frac{\pi }{2}}\left|t\right|}$$

  The behavior of the quotient is dominated by $e^{-{\displaystyle \frac{\pi }{4}}\left|t\right|}$, which suggests that $\left|V\right(s\left)\right|$ tends to zero as $\left|t\right|\to \infty$, thus ensuring boundedness in any vertical strip $1/2<a\le \mathfrak{R}\left(s\right)\le b$.

4. Limitations and Future Steps

It is imperative to acknowledge the limitations of this work and the steps that must be rigorously verified by the mathematical community.

• Rigorous Construction of $g\left(s\right)$: The definition and properties of the "corrector crutch" $g\left(s\right)$ must be formalized with extreme detail. It is necessary to demonstrate unambiguously its holomorphy and its ability to cancel all relevant analytical discrepancies in the critical region.
• Proof of Boundedness of $V\left(s\right)$: Although the asymptotic estimates are a strong indicator, a complete and detailed proof that $\left|V\right(s\left)\right|$ is bounded throughout the region $\mathfrak{R}\left(s\right)>1/2$ is required. This must consider not only the behavior for large $\left|t\right|$ but also in the vicinity of the critical line.
• Peer Review: This document presents a strategy and a sketch. Every step, every lemma, and every estimate must be subjected to the scrutiny of experts in analytic number theory. The purpose of this manuscript is precisely to invite that review.

5. Conclusion

This paper has presented a possible pathway for a proof of the Riemann Hypothesis, based on the principle of spectral transfer from the field of Gaussian integers $\mathbb{Q}\left(i\right)$ to the field of rational numbers $\mathbb{Q}$. The strategy focuses on the holomorphy of a specially designed transfer operator $V\left(s\right)$.

If it can be rigorously verified that $V\left(s\right)$ is holomorphic and bounded for $\mathfrak{R}\left(s\right)>1/2$, then the existence of a zero of the Riemann Zeta function off the critical line would lead to a mathematical contradiction. We do not claim a conclusive proof, but rather offer a path that, while technically challenging, appears logically coherent and is based on established properties of L-functions. The mathematical community is invited to analyze, critique, and, if applicable, complete the technical details outlined herein.