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\begin{center}
\vskip 1cm{\Large\bf 
On S\'andor's Inequality for the Riemann Zeta 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} 
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\begin{abstract}
Let $\omega(n)$ denote the number of distinct prime factors of $n$. We
prove an analogue of a recently published inequality of S\'andor that
relates a series involving $\omega(n)$ to a quotient of zeta functions.
\end{abstract}

\section{Introduction and statement of the main results}

The classical Riemann zeta function is defined  for real numbers  $s&gt;1$ by
$$
\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}= \prod_{p \, \text{\rm prime}} \frac{1}{1-p^{-s}}.
$$
There are several Dirichlet series whose coefficients are arithmetical functions which can be expressed in terms of the zeta function. As an example we have the
 elegant identity
\begin{equation}
\frac{\zeta^2(s)}{\zeta(2s)}=
\sum_{n=1}^\infty \frac{2^{\omega(n)}}{n^s}, \quad s&gt;1, \label{1}
\end{equation}
where
$\omega(n)$   denotes the number of distinct prime factors of  $n$; see Apostol \cite[p.\ 247]{A}, 
Hardy and Wright \cite[p.\ 265]{HW}. The  additive analogues of (\ref{1}),
$$
\zeta^2(s)+\zeta(2s)=
2\sum_{n=1}^\infty \frac{\zeta_n(s)}{n^s}, \quad s&gt;1,
$$
and
$$
\zeta^2(s)-\zeta(2s)=
2\sum_{n=2}^\infty \frac{\zeta_{n-1}(s)}{n^s}, \quad s&gt;1,
$$
where $\zeta_m(s)=\sum_{k=1}^m 1/k^s$,  are due to Hassani and Rahimpour \cite{HR}.

In 2018,  S\'andor \cite{S}  provided an  interesting counterpart of (\ref{1}), 
\begin{equation}
\sum_{n=1}^\infty \frac{\lambda^{\omega(n)}}{n^s}&lt;\frac{\zeta^\lambda(s)}{\zeta(\lambda s)}, \quad \lambda &gt;2,  \, s&gt;1. \label{2}
\end{equation}

Two questions arise naturally. Is there a similar inequality which offers an upper bound for $\zeta^{\lambda}(s)/\zeta(\lambda s)$? Do there 
exist related results for real parameters $\lambda$ with $1&lt;\lambda&lt;2$? The aim of this note is to give affirmative answers to both questions.


Our first theorem presents a  complement of (\ref{2}).


\begin{theorem} 
{For all real numbers  $\lambda&gt;2$ and $s&gt;1$, we have}
\begin{equation}
\frac{\zeta^\lambda(s)}{\zeta(\lambda s)}&lt;
\sum_{n=1}^\infty \frac{(2^{\lambda}-2)^{\omega(n)}}{n^s}. \label{3}
\end{equation}
\end{theorem}

Next, we provide upper and lower bounds for  $\zeta^{\lambda}(s)/\zeta(\lambda s)$ which are valid for all  real numbers $\lambda$ with $1&lt;\lambda&lt;2$.


\begin{theorem} 
{Let $1&lt;\lambda&lt;2$ and $s&gt;1$. Then}
\begin{equation}
\sum_{n=1}^\infty \frac{(2^{\lambda}-2)^{\omega(n)}}{n^s}&lt;
\frac{\zeta^\lambda(s)}{\zeta(\lambda s)}&lt;
\sum_{n=1}^\infty \frac{{\lambda}^{\omega(n)}}{n^s}.   \label{4}
\end{equation}
\end{theorem}







In the next section, we collect three lemmas. The proofs of the two theorems are given in Section 3.



\section{Lemmas}


A proof of the first lemma can be found in Hardy and Wright \cite[p.\ 249]{HW}.

\begin{lemma}  \label{L1}
{Let $f$ be a multiplicative arithmetical function with $f(1)=1$ such that $\sum_{n=1}^\infty f(n)/n^s$ is absolutely convergent. Then}
$$
\sum_{n=1}^\infty \frac{f(n)}{n^s} = \prod_{ p \, \text{\rm prime}} \Bigl(1+\frac{f(p)}{p^s} +\frac{f(p^2)}{p^{2s}} + \cdots\Bigr).
$$
\end{lemma}

The next lemma might be known. Since we cannot give a reference, we include a proof.



\begin{lemma} \label{L2}
{Let $a&gt;0$ and $s&gt;1$ be real numbers. The series 
\begin{equation}
\sum_{n=1}^\infty \frac{a^{\omega(n)}}{n^s}  \label{5}
\end{equation}
is convergent.}
\end{lemma}

\begin{proof}
Let $s&gt;1$.  From (\ref{1}) we conclude that if $0&lt;a\leq 2$, then the series is convergent. Next, let $a&gt;2$. A result of Robin \cite{R} states that there exists a number $b\approx 1.384$ such that
$$
\omega(n) \leq b\frac{\log n }{\log \log n }, \quad n\geq 3.
$$
This leads to
\begin{equation}
\frac{a^{\omega(n)}}{n^s} \leq \frac{a^{b\log n /\log \log n  }}{n^s}, \quad n\geq 3.   \label{6}
\end{equation}
Moreover, there exists a positive integer $m$ such that
\begin{equation}
\frac{a^{b \log n /\log \log n  }}{n^s} \leq \frac{1}{n^{(s+1)/2}}, \quad n\geq m. \label{7}
\end{equation}
From (\ref{6}) and (\ref{7}) we conclude that the series  in (\ref{5}) is convergent.
\end{proof}



\begin{lemma} \label{L3}
{Let $c$ and $x$ be real numbers.}\\
\begin{itemize}
\item[\rm (i)]  {If $1&lt;c&lt;2$ and $0&lt;x&lt;1/2$, then}
\begin{equation}
\frac{1+(2^c-3)x}{1-x}&lt;
\frac{1-x^c}{(1-x)^c}&lt; \frac{1+ (c-1)x}{1-x}.  \label{8}
\end{equation}
\item[\rm (ii)] {If $c&gt;2$ and $0&lt;x&lt;1/2$, then} {\rm (\ref{8})} {holds with} ``$&gt;$'' {instead of} ``$&lt;$''.
\end{itemize}
\end{lemma}




\begin{proof}
  Let $c&gt;0$ and $0&lt;x&lt;1$. We define
\begin{align*}
u_c(x) &amp; =\frac{1-x^c}{(1-x)^{c-1}}, \\
v_c(x) &amp; =\frac{u_c(x)-1}{x}, \\
w_c(x) &amp; =x u'_c(x)-u_c(x)+1.
\end{align*}
Then
\begin{align*}
x^2 v'_c(x) &amp; =w_c(x), \\
w'_c(x) &amp; =x u''_c(x)= \frac{ (c-1) c x (1-x^{c-2})}{(1-x)^{c+1}}.
\end{align*}

\begin{itemize}
\item[(i)] Let $1&lt;c&lt;2$ and $0&lt;x&lt;1/2$. Then $w'_c(x)&lt;0$. Thus,
$$
w_c(x)&lt;w_c(0)=0.
$$
It follows that
$$
2^{c}-3=v_{c}(1/2)&lt;v_{c}(x)&lt;v_{c}(0)=u'_{c}(0)=c-1.
$$
This implies (\ref{8}).

\item[(ii)] Let $c&gt;2$ and $0&lt;x&lt;1/2$. Then $w'_c(x)&gt;0$ and $w_c(x)&gt;w_c(0)=0$. This gives
$$
v_c(0)&lt;v_c(x)&lt;v_c(1/2)
$$
which is equivalent to (\ref{8}) with ``$&gt;$'' instead of ``$&lt;$''.
\end{itemize}
\end{proof}




\section{Proof of the theorems}

\begin{proof}
We follow the method of proof given in \cite{S}.
 Let $a&gt;0$ and $s&gt;1$. For positive integers~$n$, we define
$$
F_{a}(n)= a^{\omega(n)}.
$$
Since
$$
 \omega(1)=0 \quad\mbox{and} \quad
\omega(mn)=\omega(m)+\omega(n), \quad \gcd(m,n)=1,
$$
we obtain
$$
F_{a}(1)=1 \quad\mbox{and} \quad
F_{a}(mn)=F_{a}(m)F_{a }(n), \quad \gcd(m,n)=1.
$$
From Lemma \ref{L1} and Lemma \ref{L2}   we get
\begin{equation}
\sum_{n=1}^\infty \frac{F_{a}(n)}{n^s}=\prod_{p \, \text{\rm prime}} \Bigl(1+\frac{F_{a}(p)}{p^s}
+\frac{F_{a}(p^2)}{p^{2s}} + \cdots \Bigr).  \label{9}
\end{equation}
We have
\begin{equation}
\sum_{n=1}^\infty \frac{F_{a}(p^n)}{p^{ns}}= \sum_{n=1}^\infty \frac{ a^{\omega(p^n)}}{p^{ns}} =
\sum_{n=1}^\infty  \frac{a}{p^{ns}}= a  \frac{p^{-s}}{1-p^{-s}}. \label{10}
\end{equation}

\begin{itemize}
\item[(i)]  Let $1&lt;\lambda&lt;2$.   Using (\ref{9}), (\ref{10}) and Lemma \ref{L3}, part (i) with $x=p^{-s}$   yields
\begin{equation}
\sum_{n=1}^\infty \frac{(2^{\lambda}-2)^{\omega(n)}}{n^s}
=\prod_{p \, \text{\rm prime}} \frac{1+( 2^\lambda-3) p^{-s}}{1-p^{-s}}
&lt;
\prod_{p \, \text{\rm prime}} \frac{1- p^{-s\lambda}}{(1-p^{-s})^{\lambda}} \label{11}
\end{equation}
and
\begin{equation}
\sum_{n=1}^\infty \frac{\lambda^{\omega(n)}}{n^s}
=\prod_{p \, \text{\rm prime}} \frac{1+(\lambda-1)p^{-s}}{(1-p^{-s})^{\lambda}}
&gt;
\prod_{p \, \text{\rm prime}} \frac{1-p^{-s\lambda}}{(1-p^{-s})^{\lambda}}. \label{12}
\end{equation}
From (\ref{11}), (\ref{12}) and
\begin{equation}
\zeta^{\lambda}(s)= \prod_{p \, \text{\rm prime}}\frac{1}{(1-p^{-s})^\lambda},
\quad
\zeta(\lambda s) = \prod_{p \, \text{\rm prime}}\frac{1}{1-p^{-\lambda s}} \label{13}
\end{equation}
we conclude that (\ref{4}) is valid.


\item[(ii)] Let $\lambda&gt;2$. We apply Lemma \ref{L3}, part (ii) with $x=p^{-s}$, then we obtain (\ref{11}) with ``$&gt;$'' instead of ``$&lt;$''. Using this result and  (\ref{13}) leads to (\ref{3}).

\end{itemize}
\end{proof}




\begin{thebibliography}{9}

\bibitem{A}
T. M. Apostol, {\it Introduction to Analytic Number Theory}, Springer, 1976.

\bibitem{HW}
G. H. Hardy and E. M. Wright, {\it An Introduction to the Theory of Numbers}, 4th Ed., Oxford University Press, 1959.

\bibitem{HR}
M. Hassani and S. Rahimpour, L-Summing method, {\em RGMIA Research Report Collection} {\bf 7}(4) (2004), article 10.

\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{S}
 J. S\'andor, An inequality involving a ratio of zeta functions, {\em Notes Number Th. Disc. Math.} {\bf 24} (2018), 92--94.

\end{thebibliography}

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\noindent {2020 \it{Mathematics Subject Classification:}} Primary 11M06; Secondary 11B83, 26D15.

\noindent {\it{Keywords:}}  Riemann zeta function, prime omega function, inequality.

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\noindent (Concerned with sequence \seqnum{A129251}.)

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\vspace*{+.1in}
\noindent
Received January 8 2023;
revised versions received March 3 2023; March 5 2023.
Published in {\it Journal of Integer Sequences}, March 16 2023.

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\noindent
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