440 History of the Theory of Numbers. [Chap. XVIII 
Now Q is asjinptotic to the "integral logarithm of x": 
4=0 V/0 log U Jl+i log u/ 
so that the latter is asymptotic to irix). De la Vall^e-Poussin^^' proved that 
Lix represents t{x) more exactly than x/log x and its remaining approxi- 
mations . , ^ , 
x , a: , .{m — l)]x 
log X log^x log'"a: 
The historj^ of this extensive subject is adequately presented in the 
luminous and exhaustive text by E. Landau,"*^ in which is given (pp. 908- 
961) a complete list of references. The reader may consult the article by 
J. Hadamard,'^^ the extensive report by G. Torelli,^^^ the summaries by 
Landau,^^° also G. H. Hardy and J. E. Littlewood,^^^ and the recent papers 
42-44 of Chapter XIX. 
"'M^m. Couronn^s Acad. Roy. Belgique, 59, 1889, 1-74. 
"^Encyclopedic des sc. math., tome I, vol. 3, pp. 310-345. 
"Utti R. Accad. Sc. Fis. Mat., XapoU, (2), 11, 1902, No. 1, 222 pp. 
^oProc. Fifth Internat. Congress, Cambridge, 1, 1913, 93-108. Math. Zeitschrift, 1, 1918, 1-24, 
213-9. 
«»Acta Math., 41, 1917, 119-196. 
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