DISTRIBUTION OF THE VALUES OF THE EULER F UNCTION

Let Phi(x) denote the number of those integers n with phi(n) less than or equal to x, where phi denotes the Euler function. Improving on a w ell-known estimate of Bateman (1972), we show that Phi(x) - Ax much le ss than R(x), where A = zeta(2)zeta(3)/zeta(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.