TY - JOUR
AU - Skewes, S.
AB - ON THE DIFFERENCE TT(X)-UX (II) By S. SKEWES [Received 31 December 1953.—Read 21 January 1954] INTRODUCTION 1. LET TT(X) denote, as usual, the number of primes less than or equal to x which we suppose always to be not less than 2, and let e-0\J J \0gU 1 + e' The difference d(x) = TT(X)—]ix is negative for all values of x up to 10 , and for all the special values of x for which n(x) has been calculated (e.g. ). Littlewood (1) proved in 1914, however, d(x) = —1757 for x = 10 that d(x) changes sign infinitely often, and in particular there exists an X such tha t d(x) > 0 for some x < X. This last result is our present subject. Littlewood's method depends on an 'explicit formula', as does all subse- quent work, including the present paper. If 6 is the upper bound of the real parts of the zeros p = jS+iy of the Riemann zeta-function £(s), the 'Riemann hypothesis' [(RH) for short] is that 9 = \; if this is false, then -| < 9 < 1. In this latter case it had e e long been known that, for each 
TI - On the Difference π(x) − lix (II)
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-5.1.48
DA - 1955-03-01
UR - https://www.deepdyve.com/lp/wiley/on-the-difference-x-lix-ii-0q0ybYJ08c
SP - 48
EP - 70
VL - s3-5
IS - 1
DP - DeepDyve
ER -