On a Sum Involving the Sum-of-Divisors Function

Sφ(x) 6 π x log x + O x(log x) log2 x ( (1/3) , (4) and also proved that the error term in (4) isΩ(x), where log2 denotes the iterated logarithm. Some related works can be found in [4, 5]. Since the sum-of-divisors function σ(n) ≔ d|nd has similar properties as the Euler function φ(n) in many cases, it seems natural and interesting to consider its analogy of (3). Our result is as follows.