0 引言

对于自然数n>1, 欧拉函数φ(n)定义为不大于n且与n互素的正整数的个数.当n是素数时, 有φ(n)=n-1;当n=p₁^α₁p₂^α₂…p_k^α_k时, $\varphi \left( n \right)=\prod\limits_{i=1}^{k}{p_{i}^{{{\alpha }_{i}}-1}\left( {{p}_{i}}-1 \right)}$.特别对于素数方幂n=p^k, k∈N时, 有φ(p^k)=p^k-1(p-1).文献[1]研究了数论函数U(n), V(n)和欧拉函数φ(n)的均值分布性质, 并给出两个渐近公式, 即

$ \begin{array}{*{20}{c}} {\sum\limits_{n \le x} {\varphi \left( {{n^k}} \right)U\left( n \right)} = \frac{{{x^{k + 2}}}}{{k + 2}}\prod\limits_p {\left( {1 - \frac{2}{{p\left( {p + 1} \right)}}} \right) + O\left( {{x^{k + \frac{3}{2} + \varepsilon }}} \right)} ;}\\ {\sum\limits_{n \le x} {\varphi \left( {{n^k}} \right)U\left( n \right)} = \frac{{{x^{k + 2}}}}{{k + 2}}\prod\limits_p {\left( {1 - \frac{1}{{{p^3}}}} \right) + O\left( {{x^{k + \frac{3}{2} + \varepsilon }}} \right)} .} \end{array} $

文献[2]研究了Smarandache Ceil函数S_k(n)与欧拉函数φ(n)的均值分布性质, 并给出一个有趣的渐近公式:

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{S_k}\left( n \right)} \right)} = \frac{{6\zeta \left( {m + 1} \right)\zeta \left( {k\left( {m + 1} \right) - m} \right)}}{{{{\rm{\pi }}^2}\left( {m + 1} \right)}}R\left( {m + 1} \right){x^{m + 1}} + O\left( {{x^{k + 1/2 + \varepsilon }}} \right). $

对任意的正整数n, 伪Smarandache无平方因子函数Z_w(n)^[3]定义为最小的正整数m使得n|mⁿ, 即Z_w(n)=min{m:n|mⁿ, m∈N}.关于Z_w(n)的初等性质, 有不少学者进行研究, 并获得许多有理论价值的研究成果^[4-11].如文献[4]研究了Z_w(n)的混合均值, 并得到一个较强的渐近公式, 即

$ \sum\limits_{n \le x} {{{\left( {{Z_w}\left( n \right)} \right)}^\alpha }} = \frac{{\zeta \left( {\alpha + 1} \right){x^{\alpha + 1}}}}{{\zeta \left( 2 \right)\left( {\alpha + 1} \right)}}\prod\limits_p {\left( {1 - \frac{1}{{{p^\alpha }\left( {p + 1} \right)}}} \right) + O\left( {{x^{\alpha + 1/2 + \varepsilon }}} \right)} . $

文献[5]研究了伪Smarandache无平方因子函数Z_w(n)的混合均值问题, 并给出渐近公式，即

$ \sum\limits_{n \le x} {V\left( n \right){Z_w}\left( n \right)} = \frac{{{x^3}}}{3}\sum\limits_{i = 1}^k {\frac{{{a_i}}}{{{{\ln }^i}x}} + O\left( {\frac{{{x^3}}}{{{{\ln }^{k + 1}}x}}} \right)} , $

其中a_i(i=1, 2, …, k)为可计算的常数.

本文主要利用初等和解析的方法, 研究了欧拉函数φ(n)与伪Smarandache无平方因子函数Z_w(n)的混合均值问题, 并得到一个有趣的渐近公式.

定理1  设整数m≥1, 则对任意实数x≥1, 有

(1) 当m为偶数时,

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} = \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{{\rm{\pi }}^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 2}}}}{{{p^m}\left( {p + 1} \right)}}} \right) + O\left( {{x^{m + \frac{1}{2} + \varepsilon }}} \right)} ; $

(2) 当m为奇数时,

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} = \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{{\rm{\pi }}^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots - C_m^{m - 1}p + 2}}{{{p^m}\left( {p + 1} \right)}}} \right) + O\left( {{x^{m + 1/2 + \varepsilon }}} \right)} , $

其中ζ(s)是Riemann Zeta函数, ε为任意正实数.

1 相关引理及定义

引理1^[12]  对任意素数p和正整数k, 有Z_w(p^k)=p.

引理2^[13]  设$A\left( s \right)=\sum\limits_{n=1}^{\infty }{a\left( n \right){{n}^{-s}}}, {{\sigma }_{\alpha }}<+\infty $.再设存在递增函数H(u)及函数B(u), 使得|a(n)|≤H(n), n=1, 2, …, $\sum\limits_{n=1}^{\infty }{a\left( n \right){{n}^{-\sigma }}\le B\left( \sigma \right)}, \sigma >{{\sigma }_{\alpha }}$, 则对任意的s₀=σ₀+it₀及b₀>σ_α, 当b₀≥b>0, b₀≥σ₀+b>σ_α, T≥1及x≥1时, 有

(1) x≠正整数N时,

$ \begin{array}{l} \sum\limits_{n = 1}^\infty {a\left( n \right){n^{ - {s_0}}}} = \frac{1}{{2{\rm{\pi i}}}}\int_{b - {\rm{i}}T}^{b + {\rm{i}}T} {A\left( {{s_0} + s} \right)\frac{{{x^s}}}{s}{\rm{d}}s + O\left( {\frac{{{x^b}B\left( {b + {\sigma _0}} \right)}}{T}} \right) + O\left( {{x^{1 - {\sigma _0}}}H\left( {2x} \right)\min \left( {1,\frac{{\log x}}{T}} \right)} \right)} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;O\left( {{x^{ - {\sigma _0}}}H\left( N \right)\min \left( {1,x/T\left\| x \right\|} \right)} \right), \end{array} $

其中N是离x最近的整数(x为半奇数时, 取N=x-1/2), ‖x‖=|N-x|.

(2) x=正整数N时,

$ \begin{array}{l} \sum\limits_{n \le x} {a\left( n \right){n^{{s_0}}}} + \frac{1}{2}a\left( N \right){N^{{s_0}}} = \frac{1}{{2{\rm{\pi i}}}}\int_{b - {\rm{i}}T}^{b + {\rm{i}}T} {A\left( {{s_0} + s} \right)\frac{{{N^s}}}{s}{\rm{d}}s + O\left( {\frac{{{N^b}B\left( {B + {\sigma _0}} \right)}}{T}} \right) + } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;O\left( {{N^{1 - {\sigma _0}}}H\left( {2N} \right)\min \left( {1,\log N/T} \right)} \right). \end{array} $

这里 常数O仅和σ_a, b₀有关.

2 定理1的证明

对任意复数s(Re s>2), 设

$ f\left( s \right) = \sum\limits_{n = 1}^\infty {\frac{{{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)}}{{{n^s}}}} , $

由Euler乘积公式^[14]可得

$ \begin{array}{l} f\left( s \right) = \prod\limits_p {\left( {1 + \frac{{{\varphi ^m}\left( {{Z_w}\left( p \right)} \right)}}{{{p^s}}} + \frac{{{\varphi ^m}\left( {{Z_w}\left( {{p^2}} \right)} \right)}}{{{p^{2s}}}} + \cdots + \frac{{{\varphi ^m}\left( {{Z_w}\left( {{p^k}} \right)} \right)}}{{{p^{ks}}}} + \cdots } \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\prod\limits_p {\left( {1 + \frac{{{\varphi ^m}\left( p \right)}}{{{p^s}}} + \frac{{{\varphi ^m}\left( p \right)}}{{{p^{2s}}}} + \cdots + \frac{{{\varphi ^m}\left( p \right)}}{{{p^{ks}}}} + \cdots } \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\prod\limits_p {\left( {1 + \frac{{{{\left( {p - 1} \right)}^m}}}{{{p^s}}} + \frac{{{{\left( {p - 1} \right)}^m}}}{{{p^{2s}}}} + \cdots + \frac{{{{\left( {p - 1} \right)}^m}}}{{{p^{ks}}}} + \cdots } \right)} = \prod\limits_p {\left( {1 + \frac{{{{\left( {p - 1} \right)}^m}/{p^s}}}{{1 - 1/{p^s}}}} \right)} , \end{array} $

此处对m分情况讨论.

(1) 当m为偶数时,

$ \begin{array}{l} f\left( s \right) = \prod\limits_p {\left( {1 + \frac{{{{\left( {p - 1} \right)}^m}/{p^s}}}{{1 - 1/{p^s}}}} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\zeta \left( s \right)\prod\limits_p {\left( {1 - \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}}\left( {{p^m} - C_m^1{p^{m - 1}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 1}} + {{\left( { - 1} \right)}^m}} \right)} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\zeta \left( s \right)\prod\limits_p {\left( {1 + \frac{1}{{{p^{s - m}}}} + \frac{1}{{{p^s}}}\left( {C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 2}}} \right)} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\frac{{\zeta \left( s \right)\zeta \left( {s - m} \right)}}{{\zeta \left( {2s - 2m} \right)}}\prod\limits_p {\left( {1 - \frac{{{p^{s - m}}\left( {C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 2}}} \right)}}{{{p^s}\left( {{p^{s - m}} + 1} \right)}}} \right)} , \end{array} $

其中ζ(s)为Riemann Zeta函数, $f\left( s \right)\cdot \frac{{{x}^{s}}}{s}$在s=m+1处有一阶极点, 留数为

$ \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{{\rm{\pi }}^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 2}}}}{{{p^m}\left( {p + 1} \right)}}} \right)} . $

在Perron公式^[14]中取$b=m+\frac{3}{2}+\varepsilon $, 可得

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} = \frac{1}{{2{\rm{\pi i}}}}\int_{m + 3/2 + \varepsilon - {\rm{i}}T}^{m + 3/2 + \varepsilon + {\rm{i}}T} {f\left( s \right) \cdot \frac{{{x^s}}}{s}{\rm{d}}s + O\left( {\frac{{{x^{m + 3/2 + \varepsilon }}}}{T}} \right)} . $

将上式积分限移至$\text{Re}\ s=m+\frac{1}{2}+\varepsilon $, 并取T=x可得

$ \begin{array}{l} \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} {\rm{ = }}\frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{{\rm{\pi }}^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 2}}}}{{{p^m}\left( {p + 1} \right)}}} \right) + } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{{2{\rm{\pi i}}}}\left( {\int_{m + 3/2 + \varepsilon - {\rm{i}}T}^{m + 1/2 + \varepsilon - {\rm{i}}T} + \int_{m + 1/2 + \varepsilon - {\rm{i}}T}^{m + 1/2 + \varepsilon + {\rm{i}}T} + \int_{m + 1/2 + \varepsilon + {\rm{i}}T}^{m + 3/2 + \varepsilon + {\rm{i}}T} {} } \right)f\left( s \right) \cdot \frac{{{x^s}}}{s}{\rm{d}}s. \end{array} $

容易估计

$ \begin{array}{l} \left| {\frac{1}{{2\pi {\rm{i}}}}\int_{m + 1/2 + \varepsilon - {\rm{i}}T}^{m + 1/2 + \varepsilon + {\rm{i}}T} {f\left( s \right) \cdot \frac{{{x^s}}}{s}{\rm{d}}s} } \right| \ll \int_0^T {\left| {f\left( {m + \frac{1}{2} + \varepsilon + {\rm{i}}T} \right)} \right|\frac{{{x^{m + 1/2 + \varepsilon }}}}{{1 + \left| t \right|}}{\rm{d}}s} \ll {x^{m + 1/2 + \varepsilon }},\\ \left| {\frac{1}{{2\pi {\rm{i}}}}\left( {\int_{m + 3/2 + \varepsilon - {\rm{i}}T}^{m + 1/2 + \varepsilon - {\rm{i}}T} + \int_{m + 1/2 + \varepsilon + {\rm{i}}T}^{m + 3/2 + \varepsilon + {\rm{i}}T} {} } \right)f\left( s \right) \cdot \frac{{{x^s}}}{s}{\rm{d}}s} \right| \ll \int_{m + 1/2 + \varepsilon }^{m + 3/2 + \varepsilon } {\left| {f\left( {\sigma + {\rm{i}}T} \right)\frac{{{x^{m + 3/2 + \varepsilon }}}}{T}} \right|{\rm{d}}\sigma } \ll \frac{{{x^{m + 3/2 + \varepsilon }}}}{T} = {x^{m + 1/2 + \varepsilon }}. \end{array} $

从而当m为偶数时,

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} = \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{\pi ^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 2}}}}{{{p^m}\left( {p + 1} \right)}}} \right) + O\left( {{x^{m + 1/2 + \varepsilon }}} \right)} . $

(2) 当m为奇数时,

$ \begin{array}{l} f\left( s \right) = \prod\limits_p {\left( {1 + \frac{{{{\left( {p - 1} \right)}^m}/{p^s}}}{{1 - 1/{p^s}}}} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\zeta \left( s \right)\prod\limits_p {\left( {1 - \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}}\left( {{p^m} - C_m^1{p^{m - 1}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 1}} + {{\left( { - 1} \right)}^m}} \right)} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\zeta \left( s \right)\prod\limits_p {\left( {1 + \frac{1}{{{p^{s - m}}}} - \frac{1}{{{p^s}}}\left( {C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p + 2} \right)} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\frac{{\zeta \left( s \right)\zeta \left( {s - m} \right)}}{{\zeta \left( {2s - 2m} \right)}}\prod\limits_p {\left( {1 - \frac{{{p^{s - m}}\left( {C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots - C_m^{m - 1}p + 2} \right)}}{{{p^s}\left( {{p^{s - m}} + 1} \right)}}} \right)} , \end{array} $

其中ζ(s)为Riemann Zeta函数, $f\left( s \right)\cdot \frac{{{x}^{s}}}{s}$在s=m+1处有一阶极点, 留数为

$ \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{\pi ^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots - C_m^{m - 1}p + 2}}{{{p^m}\left( {p + 1} \right)}}} \right)} . $

同理当m为奇数时,

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} = \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{\pi ^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots - C_m^{m - 1}p + 2}}{{{p^m}\left( {p + 1} \right)}}} \right) + O\left( {{x^{m + 1/2 + \varepsilon }}} \right)} . $

综上所述，当m为偶数时,

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} = \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{\pi ^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots + C_m^{m - 1}p{{\left( { - 1} \right)}^{m - 2}}}}{{{p^m}\left( {p + 1} \right)}}} \right) + O\left( {{x^{m + 1/2 + \varepsilon }}} \right)} ; $

当m为奇数时,

$ \sum\limits_{n \le x} {{\varphi ^m}\left( {{Z_w}\left( n \right)} \right)} = \frac{{6\zeta \left( {m + 1} \right){x^{m + 1}}}}{{{\pi ^2}\left( {m + 1} \right)}}\prod\limits_p {\left( {1 - \frac{{C_m^1{p^{m - 1}} - C_m^2{p^{m - 2}} + \cdots - C_m^{m - 1}p + 2}}{{{p^m}\left( {p + 1} \right)}}} \right) + O\left( {{x^{m + 1/2 + \varepsilon }}} \right)} . $

其中ζ(s)是Riemann Zeta函数, ε为任意正实数.证毕.