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\begin{document}
\title{\VERYHuge Asal Kuvvetleri Merdiveni}
\date{}
\author{}
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%{\VERYHuge Asal Kuvvetleri Merdiveni}
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\noindent
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(144,0)
\uput{50pt}[ul]{45}(!72 72 1 ASIN 4 mul ln sub){\VERYHuge$y=\uppsi(x)$}
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\begin{gathered}\def\arraystretch{1.2}
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\text{$n$ bir asal kuvveti de\u gilse}         &0
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=\Lambda(n),\\
\sum_{n\leq x}\Lambda(n)=\uppsi(x),\text{ dolay\i s\i yla}\\
\uppsi(n)=\ln\bigl(\lcm(1,\dots,n)\bigr)
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\bm{\ln(2\uppi)=\frac{\upzeta'(0)}{\upzeta(0)}}\\
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\begin{multicols}{4}%\small
\footnotesize
B\"uy\"uk fig\"ur\"un sol alt k\"o\c sesi,
k\"u\c c\"uk \c sekildeki gibidir.
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}
$144$'ten k\"u\c c\"uk olan asal kuvvetleri
a\c sa\u g\i dad\i r.
Yukar\i da tan\i mlanan
\begin{compactitem}
\item
$\Lambda$, \textbf{von Mangoldt;}
\item
$\uppsi$, \textbf{\c Ceb\i\c sov;}
\item
$\upzeta$, \textbf{Riemann zeta;}
\item
$\Gamma$, \textbf{Gamma g\"ondermesidir.}
\end{compactitem}
Son g\"ondermenin tan\i m\i nda $s$'nin ger\c cel par\c cas\i\ pozitiftir,
ama par\c cal\i\ integrallemeyle
\begin{equation*}
\Gamma(s)=\frac{\Gamma(s+1)}s.
\end{equation*}
Analitik uzan\i mla,
pozitif olmayan tamsay\i larda hari\c c
t\"um karma\c s\i k say\i larda analitik olan $\Gamma$ g\"ondermesi elde edilir.
 
1850 \textbf{\c Ceb\i\c sov Teoremi'ne} g\"ore,
baz\i\ pozitif $A$ ve $B$ i\c cin,
b\"uy\"uk ger\c cel $x$ i\c cin
\begin{equation*}
A\leq\frac{\uppsi(x)}x\leq B.
\end{equation*}
1896 ayr\i\ ayr\i\ Hadamard'\i n ve de la Vall\'ee-Pousin'in kan\i tlad\i\u g\i\
\textbf{Asal Say\i lar Teoremi'ne} g\"ore
\begin{equation*}
  \lim_{x\to\infty}\frac{\uppsi(x)}{x}=1.
\end{equation*}
1859 \textbf{Riemann Hipotezinin} bir bi\c cimine g\"ore,
her pozitif $\epsilon$ i\c cin,
baz\i\ $A$ i\c cin,
b\"uy\"uk $x$ i\c cin
\begin{equation}\label{eqn:RH}
\left|\uppsi(x)-x\right|\leq Ax^{1/2+\epsilon}.
\end{equation}
Riemann'\i n g\"osterdi\u gine g\"ore
\begin{equation*}
\frac{\Gamma\left(\frac{1-2s}4\right)\cdot\upzeta\left(\frac{1-2s}2\right)}
{\uppi^{\textstyle\frac{1-2s}4}}=
\frac{\Gamma\left(\frac{1+2s}4\right)\cdot\upzeta\left(\frac{1+2s}2\right)}
{\uppi^{\textstyle\frac{1+2s}4}}.
\end{equation*}
Bundan dolay\i, analitik uzan\i mla,
$1$ hari\c c t\"um karma\c s\i k say\i larda analitik olan $\upzeta$ g\"ondermesi elde edilir.
G\"ondermenin
\textbf{a\c sik\^ar s\i f\i rlar\i,} negatif \c cift tamsay\i lard\i r.
Kalan s\i f\i rlar\i, $\rho$ olsun.
E\u ger $\rho=\beta+\mi\gamma$ ise, o zaman $0<\beta<1$, ve orijinal bi\c ciminde Riemann Hipotezi,
her durumda
\begin{equation}\label{eqn:RH-orig}
\beta=\frac12.
\end{equation}
1895 von Mangoldt'un kan\i tlad\i\u g\i na g\"ore
\begin{multline*}
%&\phantom{{}={}}
\lim_{h\to0^+}\frac{\uppsi(x+h)+\uppsi(x-h)}2\\
=x-\frac{\upzeta'(0)}{\zeta(0)}-\sum_{\upzeta(\omega)=0}\frac{z^{\omega}}{\omega}\\
=x-\ln(2\uppi)-\frac12\ln\left(1-\frac1{x^2}\right)-\sum_{\rho}\frac{z^{\rho}}{\rho}.
\end{multline*}
Bundan dolay\i\ Riemann Hipotezi'nin \eqref{eqn:RH} bi\c cimi,
\eqref{eqn:RH-orig} bi\c ciminden \c c\i kar.
\end{multicols}

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\tiny % \hfill \hfill \hfill
David Pierce\hfill
\url{dpierce@msgsu.edu.tr}\hfill
\url{http://mat.msgsu.edu.tr/~dpierce/}\hfill
Matematik B\"ol\"um\"u\hfill
Mimar Sinan G\"uzel Sanatlar \"Universitesi\hfill
\.Istanbul%\hfill\mbox{} 

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