几个积性函数的均值
Euler 示性函数 $\varphi(n)=n\prod_{p\mid n} \left(1-\frac1{p} \right)$ 对应的 Dirichlet 级数为 \[ \sum_{n=1}^{\infty} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}, \quad (\Re s>2), \] 交错级数对应的 Dirichlet 级数是 \[ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\varphi(n)}{n^s} = \frac{2^s-3}{2^s-1} \cdot \frac{\zeta(s-1)}{\zeta(s)} \quad (\Re s>2). \] $\varphi$ 的最佳均值估计属于 Walfisz (1963) \cite[p. 144]{Wal1963} \[ \sum_{n\leqslant x} \varphi(n) = \frac{3}{\pi^2} x^2 + O\left( x (\log x)^{2/3} (\log \log x)^{4/3} \right). \] 易得 $\varphi$ 的交错级数部分和 \[ \sum_{n\leqslant x} (-1)^{n-1} \varphi(n) = \frac1{\pi^2} x^2 + O\left( x (\log x)^{2/3} (\log \log x)^{4/3} \right). \] 1900 年 E. Landau \cite{lan} 证明了 $\varphi$ 的倒数均值为 \[ \sum_{n \leqslant x} \frac{1}{\varphi(n)} = \frac{\zeta(2) \zeta(3)}{\zeta(6)} \left( \log x + \gamma - \sum_p \frac{\log p}{p^2 - p + 1} \right) + O \left( \frac{\log x}{x} \right). \] 2013 年 Bordellès 和 Cloitre \cite[Corollary 4, (i)]{BorClo2013}, 2017 年 László Tóth \cite[Theorem 17]{László Tóth} 分别证明了 $\varphi$ 的倒数交错级数部分和公式: \[ \sum_{n \leqslant x} \frac{(-1)^n}{\varphi(n)} = \frac{\zeta(2) \zeta(3)}{3 \zeta(6)} \left( \log x + \gamma - \sum_{p} \frac{\log p}{p^2-p+1} - \frac{8 \log 2}{3} \right) + O \left( \frac{(\log x)^{5/3}}{x} \right). \] Dedekind 函数 $\psi(n)=n \prod_{p\mid n} \left(1+\frac1{p}\right)$ 对应的 Dirichlet 级数是 \[ \sum_{n=1}^{\infty} \frac{\psi(n)}{n^s} = \frac{\zeta(s)\zeta(s-1)}{\zeta(2s)} \quad (\Re s>2), \] 交错级数对应的 Dirichlet 级数是 \[ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\psi(n)}{n^s} = \frac{2^s-5}{2^s+1} \cdot \frac{\zeta(s)\zeta(s-1)}{\zeta(2s)} \quad (\Re s>2). \] $\psi$ 均值的余项最好的估计也属于 Walfisz \cite[p. 100]{Wal1963} \[ \sum_{n\leqslant x} \psi(n) = \frac{15}{2\pi^2} x^2 + O\left( x (\log x)^{2/3} \right). \] 同理可得 \[ \sum_{n\leqslant x} (-1)^{n} \psi(n) = \frac{3}{2\pi^2} x^2 + O\left( x (\log x)^{2/3} \right). \] 1979 年 Sita Ramaiah 和 Suryanarayana 研究了某些积性函数倒数的均值, 他们证明了 \cite[Corollary 4.2]{SitSur1979} \begin{align*} \sum_{n \leqslant x} \frac1{\psi(n)} & = \prod_{p\in \mathbb{P}} \left(1-\frac1{p(p+1)} \right) \left(\log x+\gamma + \sum_{p\in \mathbb{P}} \frac{\log p}{p^2+p-1} \right) \\ & \quad + O \left( x^{-1} (\log x)^{2/3} (\log \log x)^{4/3}\right). \end{align*} Bordellès 和 Cloitre \cite[Corollary 4, (iii)]{BorClo2013}, László Tóth \cite[Theorem 20]{László Tóth} 分别研究了交错级数的情形: \begin{align*} \sum_{n \leqslant x} \frac{(-1)^n}{\psi(n)} & = - \frac{1}{5} \prod_p \left( 1 - \frac{1}{p(p+1)} \right) \left( \log x + \gamma + \sum_{p} \frac{\log p}{p^2+p-1} + \frac{24 \log 2}{5} \right) \\ & \quad + O \left( \frac{(\log x)^2}{x} \right). \end{align*} 除数和函数 $\sigma(n)=\sum_{d\mid n} d$ 的 Dirichlet 级数为 \[ \sum_{n=1}^{\infty} \frac{\sigma(n)}{n^s} = \zeta(s)\zeta(s-1) \quad (\Re s>2), \] 交错级数的 Dirichlet 级数是 \[\sum_{n=1}^{\infty} (-1)^{n-1} \frac{\sigma(n)}{n^s} = \left(1-\frac{6}{2^s}+\frac{4}{2^{2s}}\right) \zeta(s)\zeta(s-1) \quad (\Re s>2). \] $\sigma$ 均值的余项最佳估计仍属于 Walfisz \cite[p. 99]{Wal1963} \[ \sum_{n\leqslant x} \sigma(n) = \frac{\pi^2}{12} x^2 + O\left( x (\log x)^{2/3} \right). \] 作为推论, 有 \[ \sum_{n\leqslant x} (-1)^{n} \sigma(n) = \frac{\pi^2}{48} x^2 + O\left( x (\log x)^{2/3} \right). \] Sita Ramaiah 和 Suryanarayana 在文章 \cite[Corollary 4.1]{SitSur1979} 中给出了 \[ \sum_{n\leqslant x} \frac1{\sigma(n)} = E \left(\log x + \gamma + F \right) + O\left( x^{-1} (\log x)^{2/3}(\log \log x)^{4/3} \right), \] 其中 \begin{align*} E =\prod_{p\in \mathbb{P}} \alpha(p), & \qquad F= \sum_{p\in \mathbb{P}} \frac{(p-1)^2 \beta(p)\log p}{p\alpha(p)}, \\ \alpha(p) = \left(1-\frac1{p} \right) \sum_{\nu=0}^{\infty} \frac1{\sigma(p^\nu)} & = 1- \frac{(p-1)^2}{p} \sum_{j=1}^{\infty} \frac1{(p^j-1)(p^{j+1}-1)}, \\ \beta(p) & = \sum_{j=1}^{\infty} \frac{j}{(p^j-1)(p^{j+1}-1)}. \end{align*} Bordellès and Cloitre \cite[Corollary 4, (v)]{BorClo2013}, László Tóth \cite[Theorem 23]{László Tóth} 分别证明了 \begin{align*} \sum_{n\leqslant x} (-1)^{n-1} \frac1{\sigma(n)} & = E\left( \left(\frac2{K} -1 \right) \left(\log x+ \gamma + F \right) +2(\log 2) \frac{K'}{K^2}\right) \\ &\quad + O\left( x^{-1} (\log x)^{5/3}(\log \log x)^{4/3} \right), \end{align*} 其中 \[ K= \sum_{j=0}^{\infty} \frac1{2^{j+1}-1}, \qquad K'= \sum_{j=1}^{\infty} \frac{j}{2^{j+1}-1}. \]
参考文献
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A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, 1963.
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E. Landau, Über die Zahlentheoretische Function φ(n) und ihre Beziehung zum Goldbachschen Satz, Nachrichten der Koniglichten Gesellschaft der Wissenschaften zu Göttingen, Mathematisch Physikalische Klasse, 1900, 177–186.
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O. Bordellès and B. Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, J. Integer Seq. 16 (2013), Article 13.6.3.
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László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, J. Integer Seq. 20 (2017), Article 17.2.1.
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V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Math. J. Okayama Univ. 21 (1979), 155–164.