76 



BELL SYSTEM TECHNICAL JOURNAL 



Thus M satisfies: 



This gives: 



M + 1 + 2"^' >n>M+2' 



n < 11 



11 < n < 20 



20 < n < 37 



37 < « < 70 



70 < n < 135 

 etc. 



M = 2 

 iW = 3 

 M = 4 

 M = 5 

 M = 6 



(5) 



3 4 5 6 7 8 9 10 11 



LOGgn 



Fig. 18 — Behaviour of ^(m). 



Our upper bound for X(;0 behaves something Hke " — with a superimposed 



saw-tooth oscillation as n varies between powers of two, due to the fact that 

 m must be an integer. If we define giji) by 



2.-M+1 _^ f^^ 



gin) 



n 



M being determined to minimize the function (i.e., M satisfying (5)), then 

 g{n) varies somewhat as shown in Fig. 18 when plotted against log2 n. The 

 maxima occur just beyond powers of two, and closer and closer to them 

 as w ^ oc . Also, the saw-tooth shape becomes more and more exact. The 

 sudden drops occur just after we change from one value of M to the next. 

 These facts lead to the following: 

 Theorem 6. (a) For all n 



\{n) < ^^—, 

 n 



{b) For almost all n 



\{n) < 



