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\headline={\ifnum\pageno<2 \hfil \else \smalltt INTEGERS: \smallrm
ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 4 (2004), \#A02
\hfil \folio \fi}

\footline={\hfil}

\centerline{\bf ON THE SYMMETRY OF DIVISOR SUMS FUNCTIONS}
\centerline{\bf IN ALMOST ALL SHORT INTERVALS}
\vskip 30pt
\centerline{{\bf G. Coppola}}
\centerline{\smallit DIIMA, University of Salerno, Fisciano, SA, ITALY}
\centerline{\tt gcoppola@diima.unisa.it}
\vskip 40pt
\centerline{\smallit Received: 4/23/03, 
Revised:  1/14/04, Accepted: 3/2/04, Published: 3/3/04 }
\vskip 30 pt

\baselineskip=5pt
\centerline{\bf Abstract}

\par
We study the symmetry of divisor sums functions $\sigma_{-s}(n) {\buildrel{def}\over{=}} \sum_{d|n}d^{-s}$ (for $\sigma = {\rm Re}(s)>0$) in almost all short intervals; by elementary methods (based on the Large Sieve) we give an exact asymptotic estimate for the mean-square
 (over $N<x\le 2N$) of their ``symmetry sum" $\sum_{|n-x|\le h}{\rm
sgn}(n-x)\sigma_{-s}(n)$ (here  ${\rm sgn}(0)=0$ and ${\rm
sgn}(t){\buildrel{def}\over{=}}t/|t|$, for $t\neq 0$).

\baselineskip=15pt
\parindent=15pt
\vskip 30pt

\line{\bf 1. Introduction and statement of the results. \hfil}
%\vskip 10 pt
\noindent
In this paper we study the "symmetry" in "almost all short intervals" of the function
$$
\sigma_{-s}(n){\buildrel{def}\over{=}} \sum_{d|n}d^{-s},
$$
\par
\noindent
where $s\in {\bf C}$ has real part $\sigma >0$.
\par
As usual, we say that something holds for "almost all" short intervals $[x-h,x+h]$, as $N < x\le 2N$, if it's true $\forall x\in]N,2N]$, with at most possibly $o(N)$ exceptions; and $[x-h,x+h]$ is "short" whenever $h=h(N)$ is increasing, $h\to \infty$ and $h=o(N)$ as $N\to \infty$.
\par
\noindent
In order to study the symmetry of distribution of $\sigma_{-s}(n)$ around $x$, as $n\in [x-h,x+h]$, we define, $\forall s\in{\bf C}$ with $\sigma >0$, the "symmetry sum"
$$
S^{\pm}(x) {\buildrel{def}\over{=}} \sum_{|n-x|\le h}  {\rm sgn}(n-x) \sum_{d|n}d^{-s}
$$
\par
\noindent
and we estimate its mean-square over the segment $N < x\le 2N$, i.e. its "symmetry integral"
$$
I_s(N,h) {\buildrel{def}\over{=}} \sum_{x\sim N}  |S^{\pm}(x)|^2.
$$
\par						% PAGE 2
\noindent
Here and hereafter \thinspace $x\sim N$ \thinspace means \thinspace $N<x\le 2N$.
\par
\noindent
The problem of the estimation of this symmetry integral has
 its origin in a paper by Kaczorowski and Perelli [KP(2)],
 where they give a conditional result for the estimate of 
the Selberg integral, i.e.
$$
J(N,h) {\buildrel{def}\over{=}} \int_{N}^{2N} \big| \sum_{x<n\le x+h}\Lambda(n)-h\big|^2 dx.
$$
\par
\noindent
(Here $\Lambda(n)$ is the von-Mangoldt function: $\Lambda(p^{\alpha})=\log p$, otherwise  $\Lambda(n)=0$.)
\par
\noindent
This integral checks the deviations, on average, of the number of primes in the short interval \thinspace $[x,x+h]$ \thinspace from its expected number; in fact, we can call PNT$([x,x+h])$ the "Prime Number Theorem" in this short interval, i.e. the estimate ${\displaystyle \sum_{x<n\le x+h}\Lambda(n)\sim h }$; actually, writing \thinspace a.a.$x\in[N,2N]$ to mean almost all, i.e. all, with at most \thinspace $o(N)$ \thinspace possible exceptions, it can be proved that
$$
J(N,h)=o(Nh^2) \Leftrightarrow \hbox{PNT}([x,x+h]) \enspace \hbox{a.a.}x\in [N,2N].
$$
\par
\noindent
The problem of PNT in almost all short intervals is very old and the actual state of the art is that we can prove it whenever $h=N^{1/6-\varepsilon(N)}$, where $\varepsilon(N)\to 0$ as $N\to \infty$, see [Z1].
\par
In passing, we remark that Zaccagnini has found, also, very important consequences of non-trivial bounds for \thinspace $J(N,h)$ \thinspace on the distribution of the zeros of the Riemann $\zeta$ function, see [Z2].
\par
The estimate given by Kaczorowski and Perelli, then, enabled them to get $J(N,h)=o(Nh^2)$ in suitable ranges (hence PNT for a.a. short intervals), conditioned to non-trivial bounds for the symmetry integral for the von Mangoldt function, i.e.
$$
I(N,h) {\buildrel{def}\over{=}} \sum_{x\sim N}  \big| \sum_{|n-x|\le h}{\rm sgn}(n-x)\Lambda(n)\big|^2
$$

\noindent
(actually, their definition of $I(N,h)$ is slightly different, but can be reduced to this one).
\par
Hence, from non-trivial bounds for $I(N,h)$ they get non-trivial bounds for $J(N,h)$ (in [KP2], Theorem 2). They prove this link by a new form of the Riemann-von Mangoldt explicit formula, see [KP1]; actually, they find that the main term of the remainders in this formula (like, also, in the classic explicit formula) contains (a form of) the symmetry sum for $\Lambda(n)$ .

\par
As the problem \thinspace of \thinspace finding \thinspace non-trivial \thinspace estimates \thinspace for \thinspace the \thinspace symmetry integral of $\Lambda$ seems hopeless, due to its apparent intractability, the author started to study other arithmetic functions; like $d(n)$, the number of divisors of \thinspace $n$, in [CS1], where by the Large Sieve the author and Salerno give asymptotics for the symmetry integral of $d(n)$.
\par						% PAGE 3
This originated the study of, also, \thinspace $\omega(n)$, the number of prime divisors of $n$, see [C1]; or even the study of the almost-all symmetry of a class of arithmetic functions, see [CS2].
\par
Also, the author studied the problem of the symmetry of primes giving estimates for the symmetry integral of averages of von Mangoldt functions (but very far from estimating the symmetry integral for \thinspace $\Lambda$, see [C2]).
\par
We hope to study, in the future, the applications of our present estimates to mean values of the Riemann zeta-function, like the moments of \enspace $\zeta(s)$.

\par
\noindent
Here we will give an asymptotic for the symmetry integral of
$$
\sigma_{-s}(n) = \sum_{d|n}d^{-s},
$$
\par
\noindent
whenever \thinspace $\sigma >0$ (if $\sigma<0$, we "flip" the divisors, like in Dirichlet hyperbola method).

\par
\noindent
For $Q=N^{1\over {2+\sigma}}$, let (hereafter $\Vert \alpha \Vert$ ${\buildrel{def}\over{=}}$ $\min_{n\in {\bf Z}}|\alpha-n|$, the distance from integers)
$$
D_s(N,h) = 2 |\zeta(1+s)|^2 N \sum_{\ell \le Q}{{\mu(\ell)}\over {\ell^{2+2\sigma}}}\sum_{k\le {Q\over {\ell}}} {1\over {k^{2\sigma}}} \left \Vert {h\over k} \right \Vert.
$$
\smallskip
\par
\noindent
Then we have (as usual, $s=\sigma + it$, $\sigma,t\in {\bf R}$), abbreviating $L{\buildrel{def}\over{=}}\log N$, the following
\smallskip
\par
\noindent {\bf Theorem 1} {\it Let} \thinspace $s\in{\bf C}$ \thinspace {\it with} \thinspace $\sigma >0$. {\it Assume that} \thinspace $h=N^\theta$, \thinspace {\it with} \thinspace $0<\theta < 1/2$ \thinspace {\it and} \thinspace $\theta < {{\sigma}\over {2+\sigma}}$. \thinspace {\it Then}
$$
I_s(N,h) = D_s(N,h) + R(N,h),
$$
\par
\noindent
{\it where}, \thinspace $\forall \sigma >0$, $R(N,h)=R(N,h,s)=o(N)$; {\it more precisely}
$$
R(N,h) \ll_s N \left( hL^2 N^{-{{\sigma}\over {\sigma+2}}} + hL N^{-{{\sigma}\over {2\sigma+4}}} h^{-\sigma} \right)\qquad \enspace \hbox{if} \enspace \sigma < 1/2;
$$
$$
R(N,h) \ll_s N \left( h L^2 N^{-{{\sigma}\over {\sigma+2}}} + \sqrt{h} L^{3/2} N^{-{{\sigma}\over {2\sigma+4}}} \right)\qquad \negthinspace \hbox{if} \enspace \sigma = 1/2;
$$
$$
R(N,h) \ll_s N \left( h L^2 N^{-{{\sigma}\over {\sigma+2}}} + \sqrt{h} L  N^{-{{\sigma}\over {2\sigma+4}}} \right)\qquad \quad \thinspace \hbox{if} \enspace \sigma > 1/2.
$$
%\smallskip
\par
\noindent
(Here the implied constant may depend on $s$, $|s|$, $\sigma$ or $t$, even on all of them.)



\par
We can also give a more explicit evaluation of the main term, by our next result, for which we need the following

\par						% PAGE 4
\noindent {\bf Definition.}
$$
\eta^{(h)}(s) {\buildrel{def}\over{=}} \sum_{n=1}^{\infty} \left \Vert {h\over n} \right \Vert n^{-s}.
$$

\par
\noindent {\bf Remark.} We explicitly remark that this series converges $\forall \sigma >0$, but to values (depending on $h$) that may grow to $\infty$, as $h\to \infty$.

\par
\noindent
We'll give the main properties of \thinspace $\eta^{(h)}(s)$ \thinspace while proving our
\smallskip
\par
\noindent {\bf Corollary 1} {\it In the same hypotheses of Theorem 1, if we suppose furthermore } $\theta<{1\over {2(\sigma+2)}}$, {\it we get }
$$
I_s(N,h) = 2 {{|\zeta(1+s)|^2}\over {\zeta(2+2\sigma)}} N \eta^{(h)}(2\sigma) + R(N,h) + {\cal O}_s\left( N\left( {1\over {h^{2\sigma}}} + {h\over {N^{{2\sigma}\over {2+\sigma}}}} \right) \right),
$$
\par
\noindent
{\it with the same bounds of Theorem 1 for } $R(N,h)$.

\par
\noindent {\bf Remark.} We emphasize that all the remainders in the Corollary are \thinspace $o(N)$, \thinspace as ensured by our hypotheses on \thinspace $h$.

\par
\noindent
The paper is organized as follows:
%\par

\noindent
%\item{---}
$\bullet$ in section 2 we give an asymptotic version of the Large Sieve;
%\item{---} 
\vskip -10pt
\noindent
$\bullet$ in section 3 we apply it to Theorem 1 and prove Corollary 1.

\vskip 30pt

\line{\bf 2. An asymptotic version of the Large Sieve. \hfil}

\noindent


\noindent {\bf Lemma 1} {\it Let} $A,B$ {\it and} $N$ {\it be natural numbers,} $M$ {\it be an integer and} \thinspace  $c_{j,d}$ \thinspace {\it be complex numbers} ($\forall j,d\in${\bf N}); {\it assume that } $a_n>0$ $\forall n\in$ {\bf N} \thinspace {\it and define}
$$
\alpha _{j,d}(x){\buildrel{def}\over{=}}\sum_{n\in {\bf N}}a_n \chi_{{\cal I}(j,d,n)}(x),
$$ 
{\it where } \thinspace ${\cal I}(j,d,n)$ \thinspace {\it is an interval
whose \thinspace endpoints \thinspace depend on these three (integer)
variables and} $\chi_{{\cal I}(j,d,n)}(x)$ {\it indicates its
characteristic function; then} 
$$
\sum_{x=M+1}^{M+N} \left| \sum_{d=A}^{B} {\sum_{j\le d}}^{*} \alpha _{j,d}(x) c_{j,d} e_d (j x) \right|^{2} = \sum_{d=A}^{B} {\sum_{j\le d}}^{*} \left| c_{j,d} \right|^{2} \sum_{x=M+1}^{M+N} \left| \alpha _{j,d}(x)\right|^2
$$
$$
+ {\cal O}\left( \alpha^2 B^2 \log B \sum_{d=A}^{B} \sum_{j\le d} \left| c_{j,d} \right|^{2} \right),
$$
{\it with} ($\alpha >0$)
$$
\alpha{\buildrel{def}\over{=}}\max_{{M<x\le M+N}\atop {j,d}} |\alpha _{j,d}(x)|\ll 1.
$$
(Here the implied constant depends at most on $A,B,M,N$).

For the proof, see [CS1] (also, compare [B]).

\vskip 30pt

\line{\bf 3. Proof of Theorem 1 and of Corollary 1. \hfil}

\noindent {\it Proof of Theorem 1.} Let \enspace ${\displaystyle \chi_q(x) {\buildrel{def}\over{=}} \sum_{{|n-x|\le h}\atop {n\equiv 0 (\bmod \thinspace q)}}{\rm sgn}(n-x) }$ \enspace and, for $h=o(\sqrt N)$,
$$
S^{\pm}_{f}(x) {\buildrel{def}\over{=}} \sum_{|n-x|\le h}  {\rm sgn}(n-x)\sum_{d|n}d^{-s} = 
$$
$$
= \sum_{|n-x|\le h}  {\rm sgn}(n-x) \sum_{{{d|n}\atop {d\le \sqrt n}}}\left(d^{-s}+\left({n\over d}\right)^{-s}\right) + {\cal O}_s \left( N^{-\sigma/2} \left( {h\over {\sqrt N}} + 1 \right) \right)
$$
$$
= \sum_{d\le \sqrt x} \left( d^{-s}\chi_d(x) + \sum_{\left| m-{x\over d}\right| \le {h\over d}} m^{-s} {\rm sgn}\left( m-{x\over d} \right) \right) + {\cal O}_s \left( {{(h/\sqrt N + 1)^2}\over {N^{{\sigma}\over 2}}} \right)
$$
$$
= \sum_{d\le \sqrt x} \left( d^{-s} + \left( {x\over d} \right) ^{-s} \right) \chi_d(x) + {\cal O}_s \left( N^{-{\sigma}/2} \right),
$$
say; changing name to the variables:
$$
\Sigma(x) {\buildrel{def}\over{=}} \sum_{q\le \sqrt x} \left(q^{-s}+\left({x\over q}\right)^{-s}\right) \chi_q(x).
$$ 
Before to apply the Large-Sieve we need to rearrange $\chi_q(x)$ exponential sum, using its Fourier coefficients property $c_{at,bt}={1\over t}c_{a,b}$ (due to the fact that \thinspace $dc_{j,d}$ \thinspace depends only upon \thinspace $j/d$; also, the mean-value $c_{d,d}$ is $0$)
$$
\chi_{q}(x)=\sum_{j<q} c_{j,q}e_q(jx)=\sum_{d|q}\sum_{{j<q}\atop {(j,q)=d}} c_{j,q}e_q(jx)= \sum_{d|q}{d\over q}\sum_{{j\le d}\atop {(j,d)=1}} c_{j,d}e_{d}(jx).
$$
Hence
$$
\Sigma(x) = \sum_{d\le \sqrt x} \left( \sum_{n\le {{\sqrt x}\over d}}{{(nd)^{-s}+(x/(nd))^{-s}}\over n}\right) {\sum_{j\le d}}^* c_{j,d} e_d(jx)
$$
$$
\qquad \qquad = \sum_{d\le \sqrt{2N}} \alpha_{d}(x) {\sum_{j\le d}}^* c_{j,d} e_d(jx), \qquad \hbox{say, where:}
$$
$$
\alpha_{d}(x) {\buildrel{def}\over{=}} 
d^{-s} \sum_{n\le {{\sqrt x}\over d}} {1\over {n^{1+s}}} + 
\left( {x\over d}\right)^{-s} \sum_{n\le {{\sqrt x}\over d}} {1\over {n^{1-s}}}.
$$
By partial summation
$$
\sum_{n\le {{\sqrt x}\over d}} {1\over {n^{1-s}}} = {{({\sqrt x}/d)^s}\over s} + {\cal O}\left( {1\over {|s|}}+1+|s|+\left( {{\sqrt x}\over d}\right)^{\sigma-1}\right);
$$
also, since $\sigma>0$,
$$
\sum_{n\le {{\sqrt x}\over d}} {1\over {n^{1+s}}} = \zeta(1+s) + {\cal O}\left( {1\over {\sigma}}\left( {d\over {\sqrt x}}\right)^{\sigma}\right),
$$
whence, uniformly for all $d\le \sqrt x$ and uniformly $\forall x\in [N,2N]$, we have 
$$
\alpha_d(x) = {{\zeta(1+s)}\over {d^s}} + {\cal O}\left( N^{-{{\sigma}\over 2}}\left[ {1\over{\sigma}} + {1\over{|s|}} + 1 + |s|\right]\right).
$$
Also, in the same range (and same uniformities) we get the bound
$$
\alpha_d(x) \ll_s d^{-\sigma} \qquad \qquad (\hbox{recall }\sigma>0).
$$
\par
In the following, the symbol ${\cal O}_s$ or, equivalently, $\ll_s$, mean a dependence on $s$ and/or on related quantities, like $|s|$, $\sigma$ or $t$.
\par
Finally, we compute (in the same uniformity ranges)
$$
|\alpha_d(x)|^2 = {{|\zeta(1+s)|^2}\over {d^{2\sigma}}} + {\cal O}_s\left( d^{-\sigma} N^{-{{\sigma}\over 2}}\right).
$$

\par
\noindent
Then, in order to use our Lemma 1, we split the range of the moduli $d$:
$$
\Sigma(x)=\Sigma_1(x)+\Sigma_2(x),
$$
say, where
$$
\Sigma_1(x) {\buildrel{def}\over{=}} \sum_{d\le Q} \alpha_{d}(x) {\sum_{j\le d}}^* c_{j,d} e_d(jx) \quad \hbox{and}
$$
$$
\Sigma_2(x) {\buildrel{def}\over{=}} \sum_{Q<d\le \sqrt{2N}} \alpha_{d}(x) {\sum_{j\le d}}^* c_{j,d} e_d(jx).
$$
By Lemma 1
$$
\sum_{x\sim N}  |\Sigma_1(x)|^2 = \sum_{d\le Q}{\sum_{j\le d}}^{*}\left|c_{j,d}\right|^{2} \sum_{x\sim N}  \left| \alpha_d(x) \right|^2 + {\cal O}\left( Q^2 L \sum_{d\le Q} \sum_{j\le d} \left| c_{j,d} \right|^{2} \right)
$$
$$
= 2\sum_{d\le Q}\sum_{\ell |d}{{\mu(\ell)}\over {\ell^2}}\left \Vert {{h\ell}\over d}\right \Vert \sum_{x\sim N}  {{|\zeta(1+s)|^2}\over {d^{2\sigma}}} + {\cal O}_s\left( N^{1-{{\sigma}\over 2}} h\sum_{d\le Q}d^{-\sigma-1} \right)
$$
$$
+ {\cal O}\left( Q^2 L \sum_{d\le Q} {h\over d} \right)
$$
$$
= D_s(N,h)+ {\cal O}_s\left( N^{1-{{\sigma}\over 2}} h \right) + {\cal O}\left( Q^2 h L^2 \right),
$$
whose main term ($D_s$ stands for "Diagonal depending on $s$") is, say, 
$$
D_s(N,h){\buildrel{def}\over{=}} 2{|\zeta(1+s)|^2} N \sum_{\ell \le Q} {{\mu(\ell)}\over {\ell^{2+2\sigma}}} \sum_{k\le {Q\over {\ell}}} {1\over {k^{2\sigma}}}\left \Vert {h\over k}\right \Vert.
$$

Here, we remark that the form in which we write the diagonal "may change", due to differences in small remainders.
\par
\noindent
Again by Lemma 1 we have (let $\alpha=Q^{-\sigma}$, this time)
$$
\sum_{x\sim N}  |\Sigma_2(x)|^2 \ll_s N \sum_{Q<d\le \sqrt{2N}} {h\over d} \left( {1\over {d^{2\sigma}}} + d^{-\sigma} N^{-\sigma /2} \right) + Q^{-\sigma}NL \sum_{Q<d\le \sqrt{2N}} {h\over d}
$$
$$
\ll_s NhQ^{-\sigma} L^2.
$$
Then, we choose $Q$ optimally, by equating the remainders of non-diagonal terms of $\sum |\Sigma_1|^2$ with these last, due to $\sum |\Sigma_2|^2$ :
$$
Q^2 h L^2 = NhQ^{-\sigma} L^2, \qquad \hbox{i.e.} \qquad Q = N^{1\over {\sigma +2}},
$$
whence
$$
\sum_{x\sim N} |\Sigma_1(x)|^2 + \sum_{x\sim N} |\Sigma_2(x)|^2 = D_s(N,h) + {\cal O}_s \left( NhL^2 N^{-{{\sigma}\over {\sigma + 2}}}\right).
$$
\par						% PAGE 8
In order to apply Cauchy inequality (to the $x$-mean of $\Sigma_1(x)\Sigma_2(x)$), we need an upper bound for $D_s$.

\noindent
It is now clear that
$$
D_s(N,h)\ll_s Nh^{1-2\sigma},\qquad \forall \sigma \in ]0, 1/2[ \thinspace;
$$
$$
D_s(N,h)\ll_s NL,\qquad \sigma = 1/2;
$$
$$
D_s(N,h)\ll_s N,\qquad \forall \sigma > 1/2.
$$
\par
By the previous estimates on the mean-squares of $\Sigma_1(x)$ and $\Sigma_2(x)$ we then get, by applying Cauchy inequality,
$$
\left| \sum_{x\sim N}  |\Sigma(x)|^2 - D_s(N,h) \right| \ll_s N \left( hL^2 N^{-{{\sigma}\over {\sigma+2}}} + hL N^{-{{\sigma}\over {2\sigma+4}}} h^{-\sigma} \right),\enspace \forall \sigma \in ]0,{1\over 2}[ \thinspace ;
$$
$$
\left| \sum_{x\sim N}  |\Sigma(x)|^2 - D_s(N,h) \right| \ll_s N \left( h L^2 N^{-{{\sigma}\over {\sigma+2}}} + \sqrt{h} L^{3/2} N^{-{{\sigma}\over {2\sigma+4}}} \right),\enspace \sigma = 1/2;
$$
$$
\left| \sum_{x\sim N}  |\Sigma(x)|^2 - D_s(N,h) \right| \ll_s N \left( h L^2 N^{-{{\sigma}\over {\sigma+2}}} + \sqrt{h} L  N^{-{{\sigma}\over {2\sigma+4}}} \right),\enspace \forall \sigma > 1/2.
$$
\par
\noindent
Finally, we get Theorem 1, by Cauchy inequality for our earlier estimate:
$$
\sum_{x\sim N}  |S^{\pm}(x)|^2 = \sum_{x\sim N}  |\Sigma(x)|^2 + {\cal O}\left( \sqrt{Nh} \sqrt{N^{1-\sigma}} \right) = D_s(N,h) + R(N,h),
$$
where $R(N,h)$ satisfies the same three bounds (one for each $\sigma$-range) as above.\quad ${\hbox{\vrule\vbox{\hrule\phantom{s}\hrule}\vrule}}$

\vskip 20pt

\line{\it Proof of Corollary 1. \hfil} We first want to render $D_s$ independent of $Q$; this is accomplished by our hypotheses on $h$, which give (in particular) $h=o(\sqrt Q)$.
\par
In fact, under this assumption we get
$$
\sum_{\ell \le Q} {{\mu(\ell)}\over {\ell^{2+2\sigma}}} 
 \sum_{k\le {Q\over {\ell}}} {1\over {k^{2\sigma}}} \left \Vert {h\over k}\right \Vert = 
\sum_{\ell \le h} {{\mu(\ell)}\over {\ell^{2+2\sigma}}} 
 \sum_{k\le {Q\over {\ell}}} {1\over {k^{2\sigma}}} \left \Vert {h\over k}\right \Vert
+ {\cal O}_s\left( h^{-2\sigma}\right)
$$
and this last sum is, by the choice $Q=N^{1\over {2+\sigma}}$,
$$
\sum_{\ell \le h} {{\mu(\ell)}\over {\ell^{2+2\sigma}}} 
 \sum_{k=1}^{\infty} {1\over {k^{2\sigma}}} \left \Vert {h\over k}\right \Vert 
+ {\cal O}_s\left( {h\over {Q^{2\sigma}}} \right) =
\qquad \qquad \qquad \qquad \qquad \qquad
$$
$$						% PAGE 9
\qquad \qquad \qquad
= \left( {1\over {\zeta(2+2\sigma)}} + {\cal O}_s\left( {1\over{h^{1+2\sigma}}} \right) \right) \eta^{(h)}(2\sigma) + {\cal O}_s\left( h N^{-{{2\sigma}\over {2+\sigma}}} \right),
$$
say, where $\forall \sigma >0$
$$
\eta^{(h)}(2\sigma) {\buildrel{def}\over{=}} \sum_{n=1}^{\infty} {1\over {n^{2\sigma}}} \left \Vert {h\over n}\right \Vert = \sum_{n\le 2h} {1\over {n^{2\sigma}}} \left \Vert {h\over n}\right \Vert + {\cal O}_s(h^{1-2\sigma}) \ll_s h.
$$

\par
\noindent
Hence, by these last estimates, we get, $\forall \sigma >0$, for $h=N^{\theta}$, $0<\theta<{1\over{2(\sigma+2)}}$
$$
\sum_{\ell \le Q} {{\mu(\ell)}\over {\ell^{2+2\sigma}}} 
 \sum_{k\le {Q\over {\ell}}} {1\over {k^{2\sigma}}} \left \Vert {h\over k}\right \Vert = 
{{\eta^{(h)}(2\sigma)}\over {\zeta(2+2\sigma)}} + {\cal O}_s\left( h^{-2\sigma} 
+ hN^{-{{2\sigma}\over {2+\sigma}}} \right)
$$
whence, in the same hypotheses,
$$
D_s(N,h) = 2{{|\zeta(1+s)|^2}\over {\zeta(2+2\sigma)}} N \eta^{(h)}(2\sigma) + {\cal O}_s\left( N {1\over {h^{2\sigma}}} + N {h\over N^{{2\sigma}\over {2+\sigma}}} \right).
$$
We explicitly remark that the remainders are $o(N)$, as $N\to \infty$, as ensured by one of our  assumptions on $h$, namely $\theta<{{\sigma}\over {2+\sigma}}$.

\par
\noindent
As an application, if $h$ is odd, we get (see above)
$$
\eta^{(h)}(2\sigma) > {1\over{2^{1+2\sigma}}}
$$
\par
\noindent
whence, as $N\to \infty$
$$
D_s(N,h) \ge {1\over{2^{2\sigma}}}{{|\zeta(1+s)|^2}\over {\zeta(2+2\sigma)}} N.
$$

\par
The previous explicit expression of \thinspace $D_s$ \thinspace proves our Corollary 1.\quad ${\hbox{\vrule\vbox{\hrule\phantom{s}\hrule}\vrule}}$

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\line{\bf References \hfil}

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\item{\bf [B]} \thinspace Bombieri, E.\thinspace - \thinspace {\sl Le Grande Crible dans la theorie analitique des nombres} \thinspace - \thinspace Asterisque 18, Societ\`{e} mathematique de France 1974. $\underline{{\tt MR\enspace 51 \thinspace \# 8057}}$
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\item{\bf [C1]} \thinspace Coppola, G. - {\sl On the symmetry of distribution of the prime-divisors function in almost all short intervals} - to appear.
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\item{\bf [C2]} \thinspace Coppola, G. - {\sl On the symmetry of primes in almost all short intervals} - to appear on Ricerche di Matematica.
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\item{\bf [CS1]} \thinspace Coppola, G. and Salerno, S. - {\sl On the symmetry of the divisor function in almost all short intervals} - to appear on Acta Arithmetica.
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\item{\bf [CS2]} \thinspace Coppola, G. and Salerno, S. - {\sl On the symmetry of arithmetical functions in almost all short intervals} - to appear.
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\item{\bf [KP1]} \thinspace Kaczorowski, J. and Perelli, A.\thinspace - \thinspace {\sl A new form of the Riemann-von Mangoldt explicit formula} - Boll. Unione Matematici Italiani B {\bf 10} (1996), no. 1, 51-66. $\underline{{\tt MR\thinspace 97g:11098}}$
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\item{\bf [KP2]} \thinspace Kaczorowski, J. and Perelli, A.\thinspace - \thinspace {\sl On the distribution of primes in short intervals} - J. Math. Soc. Japan {\bf 45} (1993), no. 3, 447-458. $\underline{{\tt MR\thinspace 94e:11100}}$
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\item{\bf [Z1]} \thinspace Zaccagnini, A.\thinspace - \thinspace {\sl Primes in almost all short intervals} - Acta \thinspace Arith. \thinspace {\bf 84} \thinspace (1998), no. 3, 225-244. \thinspace $\underline{{\tt MR\thinspace 2001k:11105}}$
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\item{\bf [Z2]} \thinspace Zaccagnini, A.\thinspace - \thinspace {\sl A conditional density theorem for the zeros \thinspace of \thinspace the \thinspace Riemann \thinspace zeta-function} - Acta \thinspace Arith. \thinspace {\bf 93} \thinspace (2000), no. 3, 293-301. \thinspace $\underline{{\tt MR\thinspace 2001k:11166}}$

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