Cauchy and Riemann sums, factorials, Ramanujan numbers and their approximations

After a short recall of the basic asymptotic relations “big O” and “small o”, we consider the Cauchy sums of the form ∑a≤k 0 ; in the case where α= 1 these are strictly related to the celebrated Riemann sums. After having learned how to approximate such sums, we apply the results to the approximation of sums of the form ∑0≤