346 ARITHMETIC 4.5.3 We might conjecture, for example, (Web site layout)
346 ARITHMETIC 4.5.3 We might conjecture, for example, that F(i) = lg($) z 0.58496; let us see how close F,( 4) comes to this value for small n. We have F (i) = 4, and F&g= f - &+;-++..- 2 f =2 ( Li+L!+… 1= 2(1 -ln2) z 0.6137; E&H = & ; ($p -& + & -L5m+2 + * * -> - 2 1 = -L+L… m2 2 3 4 C-C TTL>l 4 1 1 +… -c m>l ii ( 2m(2m + 2) -3m(3m + 2) > - = i7rz(1 -ln2)-C %, ?7L>l where S, = 1/(4m + 4) -1/(9m + 6) + 1/(16m + 
-. . . . Using the values of H, for fractional z found in Table 3 of Appendix B, we find that s1= A, S2 = 4 -In 2, s3 = # -T/(2&, etc.; a numerical evaluation yields Fz(i) ==: 0.5748. Although F1(z) = Hz, it is clear that F,(z) is difficult to calculate exactly when n is large. The distributions F,(z) were first studied by K. F. Gauss, who thought of the problem in 1800. His notebook for that year lists various recurrence relations and gives a brief table of values, including the four-place value for F2( 4) that has just been mentioned. After performing these calculations, Gauss wrote, Tam complicate evadunt, ut nulla spes superesse videatur, i.e., They come out so complicated that no hope appears to be left. Twelve years later, Gauss wrote a letter to Laplace in which he posed the problem as one he could not resolve to his satisfaction. He said, I found by very simple reasoning that, for n infinite, F,(s) = log(1 + x)/log 2. But the efforts that I made since then in my inquiries to assign F,(s) -log(1 + x)/log2 for very large but not infinite values of n were fruitless. He never published his very simple reasoning, and it is not completely clear that he had found a rigorous proof. More than 100 years went by before a proof was finally published, by R. 0. Kuz min [Atti de1 Congress0 internazionale dei matematici 6 (Bologna, 1928), 83-891, who showed that F,(z) = lg(1 + X) + O(eCAfi) for some positive constant A. The error term was improved to O(ewAn) by Paul L&y shortly afterward [Bull. Sot. Math. de fiance 57 (1929), 178-194]*; but *An exposition of L&y s interesting proof appeared in the first edition of this book.