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Full text of "History Of The Theory Of Numbers - I"

154 HlSTOEY OF THE THEOEY OF NUMBERS. [CHAP. V second formula the product equals the similar function of y' if y and y' are congruent modulo p&2. • -Pr- Consider the function [x/k] where m, n, k are positive integers and a; is a positive number. Then if S(x, k) denotes lk+2k+... +[x]*, it is proved that which for w=n = 1 becomes Sylvester's55 formula. By inversion, $m(xt n, 1) = S ju(^)^7n(n~1)/S( ~.j mn), t=l \l / where juOO is Merten's function. For k as above and k' = k/prar, ' ' x \ f y ^'n'k'}' y-o where I is the least value of j for which [x/praT+j] = 0. Hence $„,(&, n, k) can be expressed in terms of functions $m(y> n, !)• True relations are derived from the last four equations by replacing n by 1 — n and $m(x, 1 -n, &) by [a/*] Proof is given of the asymptotic formula where A is finite and independent of x, m, n, while 00 1 r p— 1 For m = w = A; = l, this result becomes that of Mertens36 (and Dirichlet21). The asymptotic expressions found for 12m(a;, n, /c) are different for the cases A set of m integers (not necessarily positive) having no common divisor > 1 is said to define a totient point. Let one coordinate, as xmt have a fixed integral value 5^0, while xl}. . ., xm_i take integral values such that [xi/xm],. . ., [xm-i/xm] have prescribed values; we obtain a compartment in space of m dimensions which contains Jm-\(xm) totient points. For example, if m = 3, rr3 = 6, and the two prescribed values are zero, there are 24 totient points (x1} x2, 6) for which OgXi<6, 0^x2<6, while xl and x2 have no common divisor dividing 6, For x{ = l or 5, x2 has 6 values; for Zi = 2 or 4, x2==l, 3 or 5; for ^ = 3, x.2 = l, 2, 5; for x,=6, x2 = 0, 1, 5. Given a closed curve r=J(0), decomposable into a finite number of segments for each of which f(Q) is a single-valued, continuous function. Let