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\begin{center}
\vskip 1cm{\LARGE\bf 
On Landau's Inequality for the \\
\vskip .1in
Prime Counting Function
}
\vskip 1cm
\large
Horst Alzer \\ 
Morsbacher Stra\ss e 10 \\  
51545 Waldbr\"ol \\ 
Germany\\
\href{mailto:h.alzer@gmx.de}{\tt h.alzer@gmx.de}\\
\ \\
Man Kam Kwong \\ 
Department of Applied Mathematics \\ 
The Hong Kong Polytechnic University \\ 
Hunghom\\
Hong Kong\\
\href{mailto:mankwong@connect.polyu.hk}{\tt mankwong@connect.polyu.hk} 
\end{center}

\vskip .2 in

\begin{abstract} 
We prove that the inequality
$
2\pi(n)-\pi(2n) \geq 2 \omega(n)
$
is valid for all $n\geq 71$. Here, $\pi(n)$ denotes the prime counting function and $\omega(n)$ denotes the number of distinct prime factors of $n$. Our inequality refines a recently published result by Zhang.
\end{abstract}


\section{Introduction and statement of the main result}


In 1909, Landau  \cite{L} conjectured that the inequality
\begin{equation}
\pi(2n)\leq 2 \pi(n)   \label{e1}
\end{equation}
is valid for all integers $n\geq 2$. Here,
 $\pi(n)$ denotes the number of primes which are less than or equal to $n$. The first proof of (\ref{e1}) was given by Rosser and Schoenfeld \cite{RS} in 1966. Moreover, they showed that the sign of equality holds in (\ref{e1}) if and only if $n\in \{2,4,10\}$.


Landau's inequality attracted the attention of several mathematicians,
who presented various extensions and counterparts of (\ref{e1}). For more
information on this subject we refer to Ehrhart \cite{E}, Mitrinovi\'c,
S\'andor, Crstici \cite[Chapter VII]{MSC}, Panaitopol \cite{P1, P2, P3},
and Vlamos \cite{V}.


Our work was inspired by an interesting paper published by Zhang \cite{Z} in 2020. He obtained a positive lower bound for the difference $2\pi(n)-\pi(2n)$. More precisely, he proved that for $n\geq 59$,
\begin{equation}
2\pi(n)-\pi(2n)>\omega(2n),  \label{e2}
\end{equation}
where $\omega(n)$ denotes the number of distinct prime factors of $n$. Here, we offer the following improvement of (\ref{e2}) for $n\geq 71$.



\begin{theorem}
Let $n\geq 71$ be an integer. Then
\begin{equation}
2\pi(n)-\pi(2n)\geq 2\omega(n), \label{e3}
\end{equation}
with equality if and only if $n\in \{78, 100, 102, 126\}$.
\label{thm1}
\end{theorem}


We note that $n=70$ is the largest integer such that (\ref{e3}) is
not true.  In the next section, we present a proof of  Theorem~\ref{thm1}
and we
show that (\ref{e3}) refines (\ref{e2}).


\section{Proof of Theorem~\ref{thm1}}

\begin{proof} 
Let
\begin{equation}
F(n)=2\pi(n)-\pi(2n)-2\omega(n).   \label{e4}
\end{equation}
We consider two cases.

\bigskip

\noindent Case 1. $71\leq n\leq 30091$.

\medskip

We used MAPLE 16 and the following computer program to verify (\ref{e3}).
\begin{quote}
\begin{verbatim}
with(NumberTheory):
w := n -> nops(PrimeFactors(n)):
F := n -> 2*pi(n)-pi(2*n)-2*w(n):
for k from 71 to 30091 do
   if F(n) <= 0 then print(k,F(n)) end if
end do;
\end{verbatim}
\end{quote}


For all $n$ we obtain that $F(n)$ is positive with exactly four exceptions:
$$
F(78)=F(100)=F(102)=F(126)=0.
$$






\noindent Case 2. $n\geq 30092$.

\medskip

We apply the  estimates
\begin{equation}
\frac{n}{\log(n)-1} \leq \pi(n) \quad (n\geq 5393),  \label{e5}
\end{equation}
\begin{equation}
\pi(n)\leq \frac{n}{\log(n)-1.1}  \quad (n\geq 60184), \label{e6}
\end{equation}
\begin{equation}
\omega(n) \leq c \frac{\log(n)}{\log(\log(n))} \quad (c=1.3841; \, n\geq 3). \label{e7}
\end{equation}
The inequalities (\ref{e5}) and (\ref{e6}) are due to Dusart \cite{D}, whereas (\ref{e7}) was proved by Robin \cite{R}. Better bounds for $\pi(n)$ were given by Berkane and Dusart \cite{BD}.

Let $F(n)$ be the function defined  in (\ref{e4}).
Using (\ref{e5}), (\ref{e6}) and (\ref{e7}) gives for $n\geq 30092$,
\begin{equation}
F(n)\geq \frac{2n}{\log(n)-1}-\frac{2n}{\log(2n)-1.1}-\frac{2c\log(n)}{\log(\log(n))}=G(n), \quad\mbox{say}. \label{e8} 
\end{equation}
We set $n=e^x$ with $x\geq \log(30092)\approx 10.31$ and $a=1.1 -\log(2)\approx 0.40$. Then,
$$
\frac{1}{2} G(n)=\frac{1}{2} G(e^x)=(1-a)\frac{e^x}{(x-1)(x-a)}-c\frac{x}{\log(x)}.
$$
Using
$$
\frac{x-y}{\log(x)-\log(y)}<\frac{x+y}{2}\quad(0<y<x),
$$
(see Mitrinovi\'c \cite[p.\ 273]{M}) with $y=1$ gives
\begin{align*}
\frac{1}{2(1-a)} (x-1)(x-a) G(e^x) & =  e^x -\frac{c}{1-a} x(x-a) \frac{x-1}{\log(x)} \\
& >  e^x -\frac{c}{1-a} x(x-a) \frac{x+1}{2}  \\
& >  1+\sum_{k=1}^5 \frac{x^k}{k! } -\frac{6}{5} x(x+1) \Bigl(x-\frac{2}{5}\Bigr) \\
& =  1+\frac{x}{600}P(x)
\end{align*}
with
$$
P(x)=5x^4+25x^3-620x^2-132x+888.
$$
Since $P$ is positive on $[10,\infty)$, we obtain $G(e^x)>0$ for $x\geq \log(30092)$. From (\ref{e8}) we conclude that $F(n)>0$ for $n\geq 30092$.
\end{proof}



Finally, we
 show that (\ref{e3}) improves Zhang's inequality (\ref{e2}).

\begin{lemma}
{For all integers $n\geq 2$, we have}
\begin{equation}
2\omega(n)\geq \omega(2n). \label{e9}
\end{equation}
{Equality holds in} {\rm (\ref{e9})} {if and only if $n=p^k$, where $p$ is an odd prime number and $k$ is a positive integer.}
\label{lemma2}
\end{lemma}

\begin{proof} Let
$$
n=\prod_{j=1}^r {p_j}^{k_j},
$$
where $p_1, \cdots, p_r$ are prime numbers with $p_1<\cdots <p_r$ and $k_1, \cdots, k_r$ are positive integers.
If  $p_1=2$, then
$$
2\omega(n)-\omega(2n)=2r-r=r>0,
$$
and if 
 $p_1>2$, then
$$ 
2\omega(n)-\omega(2n)= 2r-(r+1) =r-1 \geq 0,
$$
with equality if and only if $n={p_1}^{k_1}$.
\end{proof}


Let $n\geq 71$. If $n=p^k$, where $p\geq 3$ is a prime number and $k\geq 1$ is an integer, then $n\notin \{78, 100, 102, 126\}$. From  Theorem~\ref{thm1}
and  Lemma~\ref{lemma2} we conclude that
$$
2 \pi(n)-\pi(2n)>2\omega(n)=\omega(2n).
$$
And, if $n\neq p^k$, then
$$
2\pi(n)-\pi(2n)\geq 2\omega(n)>\omega(2n).
$$






\begin{thebibliography}{99}

\bibitem{BD}
D. Berkane and P. Dusart, On a constant relating to the prime counting function, {\em Mediterr. J. Math.} {\bf 13} (2016), 929--938.

\bibitem{D}
P. Dusart, Estimates of some functions over primes without R.H., 
arxiv preprint 	arXiv:1002.0442 [math.NT], 2010.
Available at \url{https://arxiv.org/abs/1002.0442}.

\bibitem{E}
E. Ehrhart, On prime numbers, {\em Fibonacci Quart.} {\bf 26} (1988), 271--274.

\bibitem{L}
E. Landau, {\em Handbuch der Lehre von der Verteilung der Primzahlen},
Vol.~1, Teubner, 1909.

\bibitem{M}
D. S. Mitrinovi\'c, {\em Analytic Inequalities}, Springer, 1970.

\bibitem{MSC}
D. S. Mitrinovi\'c, J. S\'andor, and B. Crstici, {\em Handbook of Number Theory, Mathematics and its Applications} {\bf 351}, Kluwer, 1996.

\bibitem{P1}
L. Panaitopol, On the inequality $p_a p_b >p_{ab}$, {\em Bull. Math. Soc. Sc. Math. Roumanie (N.S.)} {\bf 41}
 (1998), 135--139.

\bibitem{P2}
L. Panaitopol, On a theorem of Edmund Landau, {\em An. Univ. Bucure\c sti Mat.} {\bf 49} (2000), 67--72.

\bibitem{P3}
L. Panaitopol, Some generalizations for a theorem by Landau, {\em Math. Inequal. Appl.} {\bf 4} (2001), 327--330.

\bibitem{R}
G. Robin, Estimation de la fonction de Tchebychef $\theta$ sur le $k$-i\`eme nombre premier et grandes valeurs de la fonction $\omega(n)$ nombre de diviseurs premiers de $n$, {\em Acta Arith.} {\bf 42} (1983), 367--389.

\bibitem{RS}
J. B. Rosser and L. Schoenfeld, Abstracts of scientific communications,
{\em Intern. Congr. Math. Moscow}, 1966, Section 3: {\em Theory of
Numbers}.

\bibitem{V}
P. Vlamos, Properties of the function $f(x)=x/\pi(x)$, {\em
Int. J. Math. Math. Sci.} {\bf 28} (2001), 307--311.


\bibitem{Z}
S. Zhang,  An improved inequality of Rosser and Schoenfeld and its application, {\em Integers} {\bf 20} (2020), \#A103.


\end{thebibliography}



\bigskip
\hrule
\bigskip

\noindent 2020 {\it Mathematics Subject Classification}:
Primary 11A41.

\noindent \emph{Keywords: }  Landau's inequality,
prime counting function, estimate, prime omega function.


\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences 
\seqnum{A000720},
\seqnum{A001221}, and 
\seqnum{A060208}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received June 28 2022;
revised versions received  September 8 2022; September 10 2022.
Published in {\it Journal of Integer Sequences}, September 11 2022.

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