440 HISTORY or THE THEOKY OF NUMBEES. [CHAP.XVIII Now Q is asymptotic to the "integral logarithm of x": T • r / f1" du , C» du \ ita; = lim(-------h ,-----)> «-o V/o log u Ji+s log w/ so that the latter is asymptotic to v(x). De la Valle*e-Poussin317 proved that Lix represents ir(x) more exactly than a/log x and its remaining approxi- mations log x The history of this extensive subject is adequately presented in the luminous and exhaustive text by E. Landau,47 in which is given (pp. 908-961) a complete list of references. The reader may consult the article by J. Hadamard,318 the extensive report- by G. Torelli,319 the summaries by Landau,320 also G. H. Hardy and J. E. Littlewood,321 and the recent papers 42-44 of Chapter XIX. 317M6m. Couronu&i Acad. Roy. Belgique, 59, 1889, 1-74. 818Encyclop6die des sc. math., tome I, vol. 3, pp. 310-345. 819Atti R. Accad. Sc. Fis. Mat., Napoli, (2), 11, 1902, No. 1, 222 pp. 820Proc. Fifth Internat. Congress, Cambridge, 1,1913, 93-108. Math. Zeitschrift, 1, 1918,1-24, 213-9. 821 Acta Math., 41, 1917, 119-196.