A Possible Proof Of The Riemann Hypothesis
Creators: name: KAPLAN, Yunus-Emre
Publisher: name: Zenodo
Year: 2025
Subjects: Riemann Hypothesis
Asymptotic Analysis
Prime Numbers
Imbricated Quadratic Equations
Integrals
Zeta Function
Eta Function
Taylor Expansion

description: This study is my approach to the Riemann Hypothesis, which potentially provides elements for a proof of it, or even an actual proof.

This version nearly has the same content as version 104; compared to the previous version, I removed two erroneous statement I had added about the cases where $a_{+\infty} \in ]0,\frac{1}{2}[$ and $a_{+\infty} \in ]\frac{1}{2},1[$.

Before starting I define $s_n=a_n+ib_n$, a zero to the $n$-th order of truncation of $\eta$, $\eta_n$.

I start by breaking down the equality $\eta(s)=0$ (which is implied by the nullity of the continued $\zeta(s)$) by expressing $|\eta(s)|^2=0$ as $\Re(\eta(s))^2 + \Im(\eta(s))^2 = 0$, expressed with cosine and sine functions, then I use trigonometric identities to further break down the sum into a sum of squares purely relying on $a_n=\Re(s_n)$ and a double sum relying on both $a_n=\Re(s_n)$ and $b_n=\Im(s_n)$ where $s_n$ is a zero of the truncated $\eta_n(s_n)$:

$$\sum_{k=1}^{n} \frac{1}{k^{2a_n}} + 2\sum_{k=1}^{n-1} \sum_{j=k+1}^{n} \frac{(-1)^{k+j-2}\cos(b_n\ln(k/j))}{(kj)^{a_n}} \xrightarrow[n \to \infty]{} 0$$

The emergence of the sum of squares $\sum_{k=1}^{n} \frac{1}{k^{2a_n}}$ is what shows us we're on the right path, for we can compare it with $\frac{n^{1-2a_n}-1}{1-2a_n}$, and since we want to prove $\Re(s_n)=a_n\xrightarrow[n \to +\infty]{}\frac{1}{2}$, that's a very good sign to see $1-2a_n$ appear.

Using the bounded nature of the cosine function, I reformulate the equality as a problem of quadratic equations, to get rid of the dependence on $\Im(s_n)$ as well as the oscillations it induces:

$\forall x \in \mathbb{R}, -1 \le \cos(x) \le 1$ so for all $n$ sufficiently large, $ \space \exists r_n \in \mathbb{N} \space | \space -1 \le r_n \le 1$ so that:

$$ \sum_{k=1}^{n} \frac{1}{k^{2a_n}} + 2r_n\sum_{k=1}^{n-1} \sum_{j=k+1}^{n} \frac{1}{(kj)^{a_n}} \xrightarrow[n \to \infty]{} 0$$

$$\sum_{k=1}^{n} x_k^2 + 2r_n\sum_{k=1}^{n-1} \sum_{j=k+1}^{n} x_kx_j \xrightarrow[n \to \infty]{} 0, \space with \space x_1,\ldots,x_n \in \mathbb{R} \space and \space x_1=1,$$

The fact that all the $x_k$ are real requires all the corresponding $\Delta_k$ to be positive as well, which then requires:

$$\Rightarrow r_n=-\frac{1}{n+\alpha} \space with \space \alpha \in ]-3,-2[ \cup ]-2,-1]$$ (actually -3 has to be excluded I didn't pay attention to it in the paper sorry)

Then we shall substitute the $x_k$ to come back to our problem:

$$\forall k \in \{1,\ldots,n\}, \space x_k=\frac{1}{k^{a_n}}$$

It has already been proved that for all the nontrivial zeros, their real part is in $]0,1[$, therefore, using the $a_n \xrightarrow[n \to \infty]{} a_{+\infty} \in ]0,1[$ condition and asymptotic equivalences, we end up with:

$$\sum_{k=1}^{n} \frac{1}{k^{2a_n}} - \frac{1}{n}\left( \sum_{k=1}^{n} \frac{1}{k^{a_n}} \right)^2 \xrightarrow[n \to \infty]{} 0$$

I then use integrals and asymptotic analysis to observe three cases of asymptotic behaviours:



when $a_n$ converges to a real number in $]0,\frac{1}{2}[$

when $a_n$ converges to a real number in$ ]\frac{1}{2},1[$

when $a_n$ converges to $\frac{1}{2}$


Then I conclude that the only logically consistent case is when $a_n$ converges to $\frac{1}{2}$.

Actually, I find that the "real part"'s asymptotic equivalent has to be, as $n \to \infty$: $$\frac{1}{2} + o\left(\frac{1}{\ln(n)^2}\right) + ik\pi\left(\frac{\ln(n)+4-\lambda}{\ln(n)^2} + o\left(\frac{1}{\ln(n)^2}\right)\right), \space k \in \mathbb{Z}^*, \space \lambda \in [0,1]$$ and that it is therefore in fact the imaginary part that has a slower (so more prominent) rate of convergence $ik\pi\left(\frac{\ln(n)+4-\lambda}{\ln(n)^2}\right), \space k \in \mathbb{Z}^*, \space \lambda \in [0,1]$.

And this imaginary rate of convergence in that pseudo-real-part vanishes as $n \to \infty$, yielding $\frac{1}{2}$

EDIT:

- In the last months I've had some strange doubts about the constraints on $r_n$ in the system of quadratic equations, about and the realness of the $\frac{1}{k^{a_n}}$ as $n \to \infty$; but everything was actually fine (at least that's what I hope), the constraint on $r_n$ held because the $x_k$ were meant to be real, and each individual $\frac{1}{k^{a_n}}$ became real as $n \to \infty$.

- More recently, I had big doubts about the constraints on $r_n$, but this time because of careless mistakes in my drafts when I took into account any value for the squared modulo instead of 0 only.

The reasons why I have so many doubts and "dumb" questionings is because I want to rigorously stress test my statements.

NOTES:

I'll keep making changes to add additional observations or correct some statements. Soon enough, I shall make brief explanations on why the necessary conditions depending on $n$ on the zeros of the partial $\eta_n$, are valid for the zeros of the full $\eta$ as $n \to \infty$ (continuity on the critical band and uniform convergence of $\eta_n$ to $\eta$ on any subset containing the real part $[\sigma_1,\sigma_2]$ so that $0 \lt \sigma_1 \le \Re(s) \le \sigma_2 \lt 1$).

Feedback would be appreciated at: emrekaplan1978@gmail.com