We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups \(\Gamma _0(N)\) . Our main theorem unifies various results from the literature, and its significance is illustrated through the following five applications: (1) the valence formula, (2) a natural generalization of classical Rohrlich’s formula to level N, (3) an explicit version of the theorem by Bringmann–Kane–Löbrich–Ono–Rolen, (4) an extension of the generalized Rohrlich formula proposed by Bringmann–Kane, and (5) an alternative proof of the decomposition formula for twisted traces of CM values of weight 0 Eisenstein series.
