  We studied Ramanujan series $\sum_{q=1}^{\infty}G(q)c_q(a)$, where $c_q(a)$
is the well-known Ramanujan sum and the complex numbers $G(q)$, as $q\in$N, are
the Ramanujan coefficients; of course, we mean, implicitly, that the series
converges pointwise, in all natural $a$, as its partial sums $\sum_{q\le
Q}G(q)c_q(a)$ converge in C, when $Q\to \infty$. Motivated by our recent study
of infinite and finite Euler products for the Ramanujan series, in which we
assumed $G$ multiplicative, we look at a kind of (partial) smooth summations.
These are $\sum_{q\in (P)}G(q)c_q(a)$, where the indices $q$ in $(P)$ means
that all prime factors $p$ of $q$ are up to $P$ (fixed); then, we pass to the
limit over $P\to \infty$. Notice that this kind of partial sums over $P-$smooth
numbers (i.e., in $(P)$, see the above) make up an infinite sum, themselves,
$\forall P\in$P fixed, in general; however, our summands contain $c_q(a)$, that
has a vertical limit, i.e. it's supported over indices $q\in$N for which the
$p-$adic valuations of, resp., $q$ and $a$, namely $v_p(q)$, resp., $v_p(a)$
satisfy $v_p(q)\le v_p(a)+1$ and this is true $\forall p\le P$ ($P$'s fixed).
In other words, $\forall G:$N $\rightarrow$ C, here, $\sum_{q\in
(P)}G(q)c_q(a)$ is a finite sum, $\forall a\in $N, $\forall P\in $P fixed: we
will call $\sum_{q=1}^{\infty}G(q)c_q(a)$ a 'smooth Ramanujan series' if and
only if $\exists \lim_P \sum_{q\in (P)}G(q)c_q(a)\in $C, $\forall a\in $N.
Notice a very important property : smooth Ramanujan series and Ramanujan series
need not to be the same. We prove : smooth Ramanujan series converge under
Wintner Assumption. (This is not necessarily true for Ramanujan series.) We
apply this to correlations and to the Hardy--Littlewood "$2k-$Twin Primes"
Conjecture.
