On a sum involving the Euler totient function

Note on Sums Involving the Euler Function

“…To bound n≤x φ([x/n]), the main idea in Bordellès et al [3] and Wu [7] relies on an estimate of the summation…”

On a Sum Involving the Sum-of-Divisors Function

“…Very recently, Wu [2] improved (2) and Zhai [3] resolved conjecture (3) by showing S φ (x) � 6 π 2 x log x + O x(log x) (2/3) log 2 x (1/3) , (4) and also proved that the error term in (4) is Ω(x), where log 2 denotes the iterated logarithm. Some related works can be found in [4,5].…”

A variant of the prime number theorem

“…(ε, 1 2 + ε) is an exponent pair, see [7]), Bordellès' (1.3) only gives the exponent 28 59 ≈ 0.4745, which is larger than our constant 9 19 ≈ 0.4736. Some related works on the quantity (1.1) can be found in [10,15].…”