A recurrence in logarithmic Möbius sums with coprimality conditions via Euler--Maclaurin

We derive a recurrence for logarithmically weighted Möbius sums by combining Möbius inversion with the Euler–Maclaurin summation formula. Consequently, starting from estimates of orders zero, one and two, we obtain polynomial approximations for these sums for every nonnegative integer exponent of $\log$. We then extend these estimates to sums with coprimality conditions via a convolution identity and to sums weighted by the reciprocal of the sum-of-divisors function. We also establish the nonnegativity of the latter sums for every positive integer logarithmic order, proving that they never change sign.