TY - JOUR AU - Mayhew, M. J. E. AB - ON THE DIFFERENCE n(x)-]ix By A. M. COHEN and M. J . E. MAYHEW [Received 21 September 1965—Revised 4 January 1967] Introduction 1. This paper is based on ideas of A. M. Turing, as recorded in an unpublished, and in places inaccurate, manuscript. We reproduce part of Turing's introduction and, in substance, his first few lemmas, but diverge from his work in Lemma 6. 'Let v(x) be the number of primes less than x, and lia; the "logarithmic integral" of x, defined by hx = lim + h . In this, as in other matters of notation, we follow Ingham (1). We propose to investigate where TT(X) — lire is positive. If x is less than about 1-42 this quantity is positive, and from there up to 10 it is negative. The figures suggest that TT(X) — \ix ~ — a;*/loga; as x -> oo, but Littlewood (2) has proved that TT(X) — \ix changes sign infinitely often. The argument proceeds by cases, according to whether the Riemann hypothesis is true or false. In the event of the Riemann hypothesis being true Skewes (3) has shown that 10lM TT(X) - liz > 0 for some x, 2