一个算术函数的均值估计

On the Estimates of Mean Value of an Arithmetic Function

:任意自然数 n都可由若干个 1及加法、乘法运算来表示。设 f(n)为包含 1的个数最少的那种表示法中含 1的个数。已知这种算术函数平均值的上、下界估计为 3nlog3 n- 3n≤ ∑nm=1 f(m)≤ 3.81 nlog3 n+n,本文对上界估计作一点改进 ,得到 3nlog3 n- 3n≤ ∑nm =1 f(m)≤ 3.69nlog3 n+n。

For every natural number n, let f(n) be the least number of ones that can be used to represent n using ones and any number of + and × signs. The upper and lower bounds of the mean value is estimated to be 3nlog 3n-3n≤∑nm=1f(m)≤3.81nlog 3n+n. This paper gives a more accurate estimation of upper bounds, i.e.3nlog 3n-3n≤∑nm=1f(m)≤3.69nlog 3n+n, where the logs are to base 3.