  We give a simple proof of a well-known theorem of G\'al and of the recent
related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD
sums. In fact, our method obtains the asymptotically sharp constant in G\'al's
theorem, which is new. Our approach also gives a transparent explanation of the
relationship between the maximal size of the Riemann zeta function on vertical
lines and bounds on GCD sums; a point which was previously unclear. Furthermore
we obtain sharp bounds on the spectral norm of GCD matrices which settles a
question raised in [2]. We use bounds for the spectral norm to show that series
formed out of dilates of periodic functions of bounded variation converge
almost everywhere if the coefficients of the series are in $L^2 (\log\log
1/L)^{\gamma}$, with $\gamma > 2$. This was previously known with $\gamma >4$,
and is known to fail for $\gamma<2$. We also develop a sharp Carleson-Hunt-type
theorem for functions of bounded variations which settles another question
raised in [1]. Finally we obtain almost sure bounds for partial sums of dilates
of periodic functions of bounded variations improving [1]. This implies almost
sure bounds for the discrepancy of $\{n_k x\}$ with $n_k$ an arbitrary growing
sequences of integers.
