Web hosting support - 362 ARITHMETIC 4.5.3 33. [MS .!?] Let h(n) be
362 ARITHMETIC 4.5.3 33. [MS .!?] Let h(n) be the number of representations of n in the form n = xx + yy , x>y>o, x > y > 0, gcd(x, Y) = 1, integer Z, 2 , y, y , (a) Show that if the conditions are relaxed to allow Z = y , the number of repre- sentations is h(n) + ](n -1)/2j. (b) Sh ow that for fixed y > 0 and 0 < t 2 y, where gcd(t, y) = 1, and for each fixed 5 in the range 0 < 2 < n/(y + t) such that x t = n (modulo y), there is exactly one representation of n satisfying the restrictions of (a) and the condition z = t (modulo y). (c) Consequently where the sum is over all positive integers y, t, t such that gcd(t, y) = 1, t 5 y, t 5 y, tt 3 n (modulo y). (d) Show that each of the h(n) representations can be expressed uniquely in the form 3: = &rn(Zl, . . . , zm), Y = &m-1(51,. . . ,xm-l), 2 = &k(Xm+l, . . . , &n+k) d, Y =&k-1(%+2,…, %+k)d, where m, k, d, and the x3 are positive integers with x1 2 2, &+k 2 2, and d is a divisor of n. The identity of exercise 32 now implies that n/d = &m+k(xI, . . . , xmfk). Conversely, any given sequence of positive integers XI, . . , &+k such that x1 2 2, xm+k 2 2, and &m+k(xl, . . . , &+k) divides n, corresponds in this way to m + k -1 representations of n. (e) Therefore nT,, = [(5n -3)/2] + 2h(n). 34. [HM40] (H. Heilbronn.) (a) Let hd(n) be the number of representations of n as in exercise 33 such that xd < x , plus half the number of representations with xd = x . Let g(n) be the number of representations without the requirement that gcd(x, y) = 1. Prove that h(n)=c dd)g( = 2 c hd( f). z), g(n) dn dn (b) Generalizing exercise 33(b), sh ow that for d 2 1, hd(n) = C(n/(y(y k t))) + O(n), where the sum is over all integers y and t such that gcd(t,y) = 1 and 0 < t 2 y < Jnld. (c) Show that c 1 1,1,(~1(~ + t)) = P(Y) ln 2 i- 0(-l(~)), where the sum is over the range 0 < t 5 y, gcd(t,y) = 1; and where a-l(y) = c,,v(l/d). (d) Show that CIcyln cp(y)/y = zl