For positive integers n , Euler’s phi function and Dedekind’s psi function are given by $$\varphi (n) = n\prod\limits_{_{p\;\text{prime}}^{p|n}} {\left( {1 - \frac{1}{p}} \right)} \;\;\text{and}\;\;\psi (n) = n\prod\limits_{_{p\;\text{prime}}^{p|n}} {\left( {1 + \frac{1}{p}} \right)} ,$$ ( n ) = n p prime p | n ( 1 1 p ) and ( n ) = n p prime p | n ( 1 + 1 p ) , respectively. We prove that for all n ⩾ 2 we have $${\left( {1 - \frac{1}{n}} \right)^{n - 1}}{\left( {1 + \frac{1}{n}} \right)^{n + 1}} \leqslant {\left( {\frac{{\varphi (n)}}{n}} \right)^{\varphi (n)}}{\left( {\frac{{\psi (n)}}{n}} \right)^{\psi (n)}}$$ ( 1 1 n ) n 1 ( 1 + 1 n ) n + 1 ( ( n ) n ) ( n ) ( ( n ) n ) ( n ) and $${\left( {\frac{{\varphi (n)}}{n}} \right)^{\psi (n)}}{\left( {\frac{{\psi (n)}}{n}} \right)^{\varphi (n)}} \leqslant {\left( {1 - \frac{1}{n}} \right)^{n + 1}}{\left( {1 + \frac{1}{n}} \right)^{n - 1}}$$ ( ( n ) n ) ( n ) ( ( n ) n ) ( n ) ( 1 1 n ) n + 1 ( 1 + 1 n ) n 1 . The sign of equality holds if and only if n is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).

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Euler 和 Dedekind 算术函数的不等式

对于正整数 n ，欧拉的 phi 函数和 Dedekind 的 psi 函数由 $$\varphi (n) = n\prod\limits_{_{p\;\text{prime}}^{p|n}} {\left ( {1 - \frac{1}{p}} \right)} \;\;\text{and}\;\;\psi (n) = n\prod\limits_{_{p\;\text{素数}}^{p|n}} {\left( {1 + \frac{1}{p}} \right)} ,$$ ( n ) = np prime p | n ( 1 1 p ) 和 ( n ) = np prime p | n ( 1 + 1 p ) ，分别。我们证明对于所有 n ⩾ 2 我们有 $${\left( {1 - \frac{1}{n}} \right)^{n - 1}}{\left( {1 + \frac{1} {n}} \right)^{n + 1}} \leqslant {\left( {\frac{{\varphi (n)}}{n}} \right)^{\varphi (n)}}{\ left( {\frac{{\psi (n)}}{n}} \right)^{\psi (n)}}$$ ( 1 1 n ) n 1 ( 1 + 1 n ) n + 1 ( ( n ) n ) ( n ) ( ( n ) n ) ( n ) 和 $${\left( {\frac{{\varphi (n)}}{n}} \right)^{\psi (n)} }{\left( {\frac{{\psi (n)}}{n}} \right)^{\varphi (n)}} \leqslant {\left( {1 - \frac{1}{n} } \right)^{n + 1}}{\left( {1 + \frac{1}{n}} \right)^{n - 1}}$$ ( ( n ) n ) ( n ) ( ( n ) n) ( n ) ( 1 1 n ) n + 1 ( 1 + 1 n ) n 1 。等号成立当且仅当 n 是素数。第一个不等式改进了 Atanassov (2011) 和 Kannan & Srikanth (2013) 的结果。等号成立当且仅当 n 是素数。第一个不等式改进了 Atanassov (2011) 和 Kannan & Srikanth (2013) 的结果。等号成立当且仅当 n 是素数。第一个不等式改进了 Atanassov (2011) 和 Kannan & Srikanth (2013) 的结果。