The Riemann Hypothesis and the possible proof
Jin Gyu Lee\medskip
{\normalsize Department of Electrical Engineering and Computer Science
The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the application of the Riemann Hypothesis will be certified, e.g. research on prime numbers.
1402.2822.tex
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\begin{document}

\title{The Riemann Hypothesis and the possible proof}
\author{Jin Gyu Lee\medskip\\{\normalsize Department of Electrical Engineering and Computer Science}\\{\normalsize Seoul National University, Seoul 151-744, Korea}}
\date{February 9, 2014}
\maketitle

\begin{abstract}
The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the application of the Riemann Hypothesis will be certified, e.g. research on prime numbers.
\end{abstract}

\section{Introduction}

 Yet the Riemann Hypothesis still remains as hypothesis, there has been a numerous effort to prove the Riemann Hypothesis until now. 
 
 This paper focuses on the examination of the possible proof for the Riemann Hypothesis. Proof of the Riemann Hypothesis will certify various applications of the hypothesis and will accelerate studies about prime numbers \cite{r1} and \cite{r2}.
 
 The article mainly deals with the region in complex plane which has its real part between 0 and 1. By the former proofs about other complex regions we can finalize the proof of the Riemann Hypothesis. Section 2 begins by defining some notations and providing previous results. It will then go on to the main result which is in section 3 and containing some lemmas and theorems about the proof of the Riemann Hypothesis, and the full paper is organized in section 4.

\section{Preliminaries}

\noindent {\bf Definition 2.1. ((1.1) of \cite{r3})} {\it The Riemann zeta-function $\zeta(s)$ is defined as
\begin{equation*}
\zeta(s)= \sum_{n=1}^{\infty} {\frac{1}{n^s}} \text{    } (Re(s)>1),
\end{equation*}
and the series converges absolutely and uniformly in the region.}\\ \\

\noindent {\bf Theorem 2.2. (Theorem 1.2 of \cite{r3})} {\it The function $\zeta(s)$, defined in Definition 2.1 for $Re(s) > 1$, admits of analytic continuation over the open right half complex plane except $s = 1$ and has the form
\begin{equation*}
\zeta(s) = (1-2^{1-s})^{-1}\sum_{n=1}^{\infty} {\frac{(-1)^{n+1}}{n^{s}}} \text{    } (Re(s)>0).
\end{equation*}}\\ \\

\noindent {\bf Theorem 2.3. (Theorem 1.6 of \cite{r3})} {\it For all complex $s$
\begin{equation*}
\zeta(s) = 2^s\pi^{s-1}\sin(\frac{s\pi}{2})\Gamma(1-s)\zeta(1-s).
\end{equation*}}\\ \\

\noindent {\bf Theorem 2.4.} {\it $Re(s) \geq 1$ is the regions in the complex plane where $\zeta(s) \neq 0$.} \\ \\

\noindent {\bf Proof.} Proof is clear by the quote ''This shows that $\zeta(s) \neq 0$ for $Re(s) > 1$'' \cite{r3}, and the Theorem 1.5 of \cite{r3}.\\ \\

\noindent {\bf Theorem 2.5.} {\it The Riemann zeta-function $\zeta(s)$, has no non-trivial zeros in the regions $Re(s) \leq 0$, where trivial zeros mean negative even numbers.} \\ \\

\noindent {\bf Proof.} According to Theorem 2.3 and Theorem 2.4, the Riemann zeta-function becomes zero in the region $Re(s) \leq 0$ only when $\sin(\frac{s\pi}{2})$ is zero, because $\Gamma(1-s)$ and $\zeta(1-s)$ has no zero when $Re(1-s) \geq 1$. \\ \\



\section{Proof of the Riemann Hypothesis}

\noindent {\bf Definition 3.1.}{\it The function $\Omega(n)$ represents the total number of prime factors of $n$.} \\ \\

\noindent {\bf Definition 3.2.} {\it We define a function $\beta(n)$ of natural number n as
\begin{equation*}
\beta(n) = \sum_{\forall m \mid n}^{} {(-1)^{\frac{n}{m}+1}(-1)^{\Omega(m)}}.
\end{equation*}} \\ \\

\noindent {\bf Theorem 3.3.} {\it The function $\beta(n)$ defined in Definition 3.2 has the following values
\begin{equation*}
\beta(n)=
\begin{cases}
1&\text{if } n = m^2, \text{ where } m\in\mathbb{N},\\
-2&\text{if } n = 2m^2, \text{ where } m\in\mathbb{N},\\
0&\text{otherwise}.
\end{cases}
\end{equation*}} \\ \\

\noindent {\bf Proof.} By the Definition 3.2 with an odd prime number $p$ and a number $n$
\begin{equation*}
\beta(p^n)=
\begin{cases}
1&\text{if } \text{n is an even number},\\
0&\text{otherwise}.
\end{cases}
\end{equation*}

Moreover for an odd prime number $p$ and an odd number $m$ satisfying $p \nmid m$ with an arbitrary natural number $a$, the beta function satisfies
\begin{align*}
\beta(p^am) = \sum_{\forall t \mid p^am}^{} {(-1)^{\frac{p^am}{t}+1}(-1)^{\Omega(t)}} = \sum_{i=0}^{a} {\sum_{\forall t \mid m}^{} {(-1)^{\frac{p^am}{p^it}+1}(-1)^{\Omega(p^it)}}} \\ = \sum_{i=0}^{a} {\sum_{\forall t \mid m}^{} {(-1)^{p^{a-i}\frac{m}{t}+1}(-1)^{\Omega(p^i)}(-1)^{\Omega(t)}}} = \sum_{i=0}^{a} {(-1)^i\sum_{\forall t \mid m}^{} {(-1)^{p^{a-i}\frac{m}{t}+1}(-1)^{\Omega(t)}}} \\ = \sum_{i=0}^{a} {(-1)^i\sum_{\forall t \mid m}^{} {(-1)^{\frac{m}{t}+1}(-1)^{\Omega(t)}}} = \sum_{i=0}^{a} {(-1)^i\beta(m)} = \beta(m)\sum_{i=0}^{a} {(-1)^i} \\  = \beta(m)\sum_{i=0}^{a} {(-1)^{\frac{p^a}{p^i}+1}(-1)^i} = \beta(m)\beta(p^a). 
\end{align*}

Using the proven knowledge, $\beta(n) = 1$ for odd square number $n$, and $\beta(n) = 0$ for odd non-square number $n$. Since even numbers can be decomposed to $2^{a}m$ with an odd number {\it m}, we have the following equation.
\begin{align*}
\beta(2^am) = \sum_{\forall t \mid 2^am}^{} {(-1)^{\frac{2^am}{t}+1}(-1)^{\Omega(t)}} = \sum_{i=0}^{a} {\sum_{\forall t \mid m}^{} {(-1)^{\frac{2^am}{2^it}+1}(-1)^{\Omega(2^it)}}} \\ = \sum_{i=0}^{a} {\sum_{\forall t \mid m}^{} {(-1)^{2^{a-i}\frac{m}{t}+1}(-1)^{\Omega(2^i)}(-1)^{\Omega(t)}}} = \sum_{i=0}^{a} {(-1)^i\sum_{\forall t \mid m}^{} {(-1)^{2^{a-i}\frac{m}{t}+1}(-1)^{\Omega(t)}}} \\ = -\sum_{i=0}^{a-1} {(-1)^i\sum_{\forall t \mid m}^{} {(-1)^{\frac{m}{t}+1}(-1)^{\Omega(t)}}} + (-1)^a\sum_{\forall t \mid m}^{} {(-1)^{\frac{m}{t}+1}(-1)^{\Omega(t)}} \\ = -\sum_{i=0}^{a-1} {(-1)^i\beta(m)} + \beta(m)(-1)^a.
\end{align*}

If {\it m} is not a square number, then $\beta(2^am)=0$ by using the fact derived above. If {\it m} is a square number, the following result holds and thus the theorem is proved.
\begin{equation*}
\beta(2^am)=
\begin{cases}
1&\text{if $a$ is even},\\
-2&\text{if $a$ is odd}.
\end{cases}
\end{equation*} \\ \\

\noindent {\bf Definition 3.4.} {\it We define a double sequence $a_{ij}$ for all $i,j \in \mathcal{N}$ as
\begin{equation*}
a_{ij} =
\begin{cases}
\frac{(-1)^{\frac{j}{i}+1}}{j^{s}}(-1)^{\Omega(i)}&\text{if $i \mid j$},\\
0&\text{otherwise},
\end{cases}
\end{equation*}
where $s \in \mathcal{C}$ satisfies $\sum_{n=1}^{\infty} {\frac{(-1)^{n+1}}{n^{s}}} = 0$ and $\frac{1}{2} < Re(s) < 1$.} \\ \\

\noindent {\bf Lemma 3.5.} {\it The double sequence defined in Definition 3.4 satisfy
\begin{equation*}
\sum_{i=1}^{\infty} {\sum_{j=1}^{\infty} {a_{ij}}} = \sum_{j=1}^{\infty} {\sum_{i=1}^{\infty} {a_{ij}}}.
\end{equation*}} \\ \\

\noindent {\bf Proof.} By Theorem 2.15 and Theorem 2.13 of \cite{r4} it is sufficient to prove 1) $\sum_{j=1}^{\infty} {a_{ij}}$ converges uniformly on $i$, 2) $\sum_{i=1}^{\infty} {\sum_{j=1}^{\infty} {a_{ij}}} = 0$, and 3) $\sum_{i=1}^{\infty} {a_{ij}}$ converges for all $j$, where 2) and 3) is eaily proved by using Definition 3.4 and the fact that $\sum_{j=1}^{\infty} {a_{ij}} = \sum_{l=1}^{\infty} {a_{i(il)}}$ for all $i$.
\begin{equation*}
\sum_{i=1}^{\infty} {\sum_{j=1}^{\infty} {a_{ij}}} = \sum_{i=1}^{\infty} {\sum_{l=1}^{\infty} {a_{i(il)}}} = \sum_{i=1}^{\infty} {\frac{(-1)^{\Omega(i)}}{i^{s}}(\sum_{l=1}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}})} = 0,
\end{equation*}
\begin{equation*}
\sum_{i=1}^{\infty} {a_{ij}} = \sum_{\forall i \mid j} {\frac{(-1)^{\frac{j}{i}+1}(-1)^{\Omega(i)}}{j^{s}}} = \frac{\beta(j)}{j^{s}}, \text{ } \forall j.
\end{equation*}

For 1), since $s$ satisfy $\sum_{l=1}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}} = 0$, there exists a $M$ s.t. for all $0 \leq n$ it satisfies $|\sum_{l=n+1}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}}| = |\sum_{l=1}^{n} {\frac{(-1)^{l+1}}{l^{s}}}| < M$. Moreover, $0 < \forall \epsilon$, $2 \leq \exists N_{1}(\epsilon)$ s.t. $N_{1}(\epsilon) \leq \forall n$ satisfies $|\sum_{l=n}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}}| = |\sum_{l=1}^{n-1} {\frac{(-1)^{l+1}}{l^{s}}}| < \epsilon$. Since, $M$ defined eariler is a constant, for all $0 < \epsilon$ there exists $N_{2}(\epsilon)$ s.t. for all $N_{2}(\epsilon) \leq m$ we have $|\frac{M}{m^{s}}|<\epsilon$.

Finally, for all $0 < \epsilon$ if we let $N(\epsilon) = N_{1}(\epsilon)*N_{2}(\epsilon)$, then for all $N(\epsilon) < n_{1}$ it satisfies $|\sum_{n=1}^{n_{1}-1} {a_{mn}}|=|\sum_{n=n_{1}}^{\infty} {a_{mn}}|<\epsilon$ for all $m$. This is true since, if $N_{2}(\epsilon) \leq m$
\begin{equation*}
|\sum_{n=n_{1}}^{\infty} {a_{mn}}| = |\frac{(-1)^{\Omega(m)}}{m^{s}}\sum_{l=[\frac{n_{1}}{m}]+1}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}}| \leq |\frac{1}{m^{s}}||\sum_{l=[\frac{n_{1}}{m}]+1}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}}| < \frac{M}{|m^{s}|} < \epsilon.
\end{equation*}

If $1 \leq m < N_{2}(\epsilon)$, then $N_{1}(\epsilon) \leq [\frac{n_{1}}{m}]+1$ holds and 
\begin{equation*}
|\sum_{n=n_{1}}^{\infty} {a_{mn}}| = |\frac{(-1)^{\Omega(m)}}{m^{s}}\sum_{l=[\frac{n_{1}}{m}]+1}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}}| \leq |\frac{1}{m^{s}}||\sum_{l=[\frac{n_{1}}{m}]+1}^{\infty} {\frac{(-1)^{l+1}}{l^{s}}}| < \frac{\epsilon}{|m^{s}|} \leq \epsilon.
\end{equation*}

Which proves the uniform convergence of $\sum_{j=1}^{\infty} {a_{ij}}$ to $0$ and proves the result.\\ \\

\noindent {\bf Theorem 3.6.} {\it The Riemann zeta-function only has zeros with real value 1/2 in the regions $0 < Re(s) < 1$.} \\ \\

\noindent {\bf Proof.} WLOG we can find $s \in \mathcal{C}$ s.t. $\frac{1}{2} < Re(s) < 1$ and $\zeta(s) = 0$ by the functional equation proved in Theorem 2.3, i.e., $\zeta(s)=(1-2^{1-s})^{-1}\sum_{n=1}^{\infty} {\frac{(-1)^{n+1}}{n^{s}}}=0$, which concludes $\sum_{n=1}^{\infty} {\frac{(-1)^{n+1}}{n^{s}}}=0$, because $(1-2^{1-s})^{-1} \neq 0$. Thereby multiplying $\frac{(-1)^{\Omega(m)}}{m^{s}}$ to the equation, $\sum_{n=1}^{\infty} {\frac{(-1)^{n+1}(-1)^{\Omega(m)}}{(mn)^{s}}}=0$ for all $m$. Adding it from $1$ to $\infty$, $\sum_{m=1}^{\infty} {\sum_{n=1}^{\infty} {\frac{(-1)^{n+1}(-1)^{\Omega(m)}}{(mn)^{s}}}=0}$ is true, which results into $0 = \sum_{i=1}^{\infty} {\sum_{j=1}^{\infty} {a_{ij}}} = \sum_{j=1}^{\infty} {\sum_{i=1}^{\infty} {a_{ij}}} = \sum_{j=1}^{\infty} {\frac{\beta(j)}{j^{s}}}$ by Lemma 3.5.

Since, $\frac{1}{2} < Re(s)$ the value of $\sum_{n=1}^{\infty} {\frac{1}{n^{2s}}}$ is defined and equals to $\zeta(2s)$. Therefore, if we define a sequence $\{b_{n}\}$ and $\{c_{n}\}$ as
\begin{equation*}
b_{n} =
\begin{cases}
\frac{1}{n^{s}}&\text{if } n = m^2, \text{ where } m\in\mathbb{N},\\
0&\text{otherwise},
\end{cases}
\end{equation*}
\begin{equation*}
c_{n} =
\begin{cases}
\frac{-2}{n^{s}}&\text{if } n = 2m^2, \text{ where } m\in\mathbb{N},\\
0&\text{otherwise}.
\end{cases}
\end{equation*}

Each series $\sum_{n=1}^{\infty} {b_{n}}$ and $\sum_{n=1}^{\infty} {c_{n}}$ converges to $\zeta(2s)$ and $-2^{1-s}\zeta(2s)$ respectively. By Lemma 3.3, $\sum_{j=1}^{\infty} {\frac{\beta(j)}{j^{s}}} = \sum_{j=1}^{\infty} {b_{j}} + \sum_{j=1}^{\infty} {c_{j}}$ and thus $0 = \sum_{j=1}^{\infty} {\frac{\beta(j)}{j^{s}}} = (1-2^{1-s})\zeta(2s)$, which concludes $\zeta(2s) = 0$.

Above statement proves that $2s$ is also a zero of the Riemann zeta-function. However, we already know by Theorem 2.4 that there is no zero for the region $Re(s) \geq 1$. Thus we have a contradiction and the theorem is proved.  \\ \\

\section{Conclusion}

For an unproved Millennium problem, Riemann Hypothesis, we give a possible proof and therefore, the use of the Riemann Hypothesis will be furtilized. \\ \\

\begin{thebibliography}{99}

\bibitem{r1} E. Landau, {\it Primzahlen}, Zwei Bd., IInd ed., with an Appendix by Dr. Paul T. Bateman, Chelsea, New York 1953.

\bibitem{r2} H.M. Edwards, {\it Riemann’s Zeta Function}, Academic Press, New York - London 1974.

\bibitem{r3} A. Ivic, {\it The Riemann Zeta-Function: Theory and Applications}, Courier Dover Publication, 2003.

\bibitem{r4} Habil, Eissa D. {\it Double sequences and double series}, The Islamic University Journal, Series of Natural Studies and Engineering 14, 2006.


\end{thebibliography}

\end{document}
1402.2822.extracted_bib
{r1 E. Landau, {Primzahlen, Zwei Bd., IInd ed., with an Appendix by Dr. Paul T. Bateman, Chelsea, New York 1953.
1402.2822.extracted_bib
{r2 H.M. Edwards, {Riemann’s Zeta Function, Academic Press, New York - London 1974.
1402.2822.extracted_bib
{r3 A. Ivic, {The Riemann Zeta-Function: Theory and Applications, Courier Dover Publication, 2003.
1402.2822.extracted_bib
{r4 Habil, Eissa D. {Double sequences and double series, The Islamic University Journal, Series of Natural Studies and Engineering 14, 2006.