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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-
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.