Weighted integrability of sums of series with respect to multiplicative systems

A necessary and sufficient condition for $L^p$-integrability with power weight of a function $f$ represented by the series with respect to multiplicative systems with generalized monotone coefficients is obtained. The integrability of the majorant of partial sums of a representing series is also described by the same conditions. In addition we study the integrability of difference quotient $(f(x)-f(0))/x$.