Series in Multiplicative Systems in Lorentz Spaces

Series in multiplicative systems $\chi$ with generalized monotone coefficients are studied. Necessary and sufficient Hardy–Littlewood type conditions for the sums of such series to belong to the Lorentz space are proved. As corollaries, we establish estimates of best approximation in the system $\chi$ and Konyushkov-type theorems on the equivalence of $O$- and $\asymp$-relations for the weighted sums of the Fourier coefficients in the system $\chi$ and for the best approximations.