  Given a periodic function $f$, we study the convergence almost everywhere and
in norm of the series $\sum_{k} c_k f(kx)$. Let $f(x)= \sum_{m=1}^\infty a_m
\sin {2\pi m x}$ where $\sum_{m=1}^\infty a_{m }^2d(m) <\infty$ and
$d(m)=\sum_{d|m} 1$, and let $f_n(x) = f(nx)$. We show by using a new
decomposition of squared sums that for any $K\subset \N$ finite, $ \|\sum_{k\in
K} c_k f_k \|_2^2 \le ( \sum_{m=1}^\infty a_{m }^2 d(m)
  ) \sum_{k\in K } c_{k}^2d(k^2)$. If $f^s (x)= \sum_{j=1}^\infty \frac{\sin
2\pi jx}{j^s}$,
  $s>1/2$, by only using
  elementary Dirichlet convolution calculus, we show that for $0< \e\le 2s-1$,
$\zeta(2s)^{-1} \|\sum_{k\in K} c_k f^s_k \|_2^2 \le \frac{1+\e}{\e } (\sum_{k
\in K}
  |c_k|^2 \s_{ 1+\e-2s}(k) )$, where $\s_h(n)=\sum_{d|n}d^h$. From this we
deduce that if $f\in {\rm BV}(\T)$, $\langle f,1\rangle=0$ and
  $$\sum_{k} c_k^2\frac{(\log\log k)^4}{(\log\log \log k)^2} <\infty,$$ then
the series $ \sum_{k } c_kf_k$ converges almost everywhere.
  This slightly improves a recent result, depending on a fine analysis on the
polydisc (\cite{ABS}, th.3) ($n_k=k$), where it was assumed that $ \sum_{k}
c_k^2 \, (\log\log k)^\g $ converges for some $\g>4$. We further show that the
same conclusion holds under the arithmetical condition $$\sum_{ k }
  c_k^2 (\log\log k)^{2 + b} \s_{ -1+\frac{1}{(\log\log k)^{ b/3}} }(k)
<\infty,$$ for some $b>0$, or if $ \sum_{ k} c_k^2 d(k^2) (\log\log k)^2
<\infty$. We also derive from a recent result of Hilberdink an $\O$-result for
the Riemann Zeta function involving factor closed sets.
  We finally prove an important complementary result to Wintner's famous
characterization of mean convergence of series $\sum_{k=0}^\infty c_k f_k $.

Number Theory