幾個積性函數的均值

Euler 示性函數 $\varphi(n)=n\prod_{p\mid n} \left(1-\frac1{p} \right)$ 對應的 Dirichlet 級數為 \[ \sum_{n=1}^{\infty} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}, \quad (\Re s>2), \] 交錯級數對應的 Dirichlet 級數是 \[ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\varphi(n)}{n^s} = \frac{2^s-3}{2^s-1} \cdot \frac{\zeta(s-1)}{\zeta(s)} \quad (\Re s>2). \] $\varphi$ 的最佳均值估計屬于 Walfisz (1963) [1, p. 144] \[ \sum_{n\leqslant x} \varphi(n) = \frac{3}{\pi^2} x^2 + O\left( x (\log x)^{2/3} (\log \log x)^{4/3} \right). \] 易得 $\varphi$ 的交錯級數部分和 \[ \sum_{n\leqslant x} (-1)^{n-1} \varphi(n) = \frac1{\pi^2} x^2 + O\left( x (\log x)^{2/3} (\log \log x)^{4/3} \right). \] 1900 年 E. Landau [2] 證明了 $\varphi$ 的倒數均值為 \[ \sum_{n \leqslant x} \frac{1}{\varphi(n)} = \frac{\zeta(2) \zeta(3)}{\zeta(6)} \left( \log x + \gamma - \sum_p \frac{\log p}{p^2 - p + 1} \right) + O \left( \frac{\log x}{x} \right). \] 2013 年 Bordellès 和 Cloitre [3, Corollary 4, (i)], 2017 年 László Tóth [4, Theorem 17] 分別證明了 $\varphi$ 的倒數交錯級數部分和公式: \[ \sum_{n \leqslant x} \frac{(-1)^n}{\varphi(n)} = \frac{\zeta(2) \zeta(3)}{3 \zeta(6)} \left( \log x + \gamma - \sum_{p} \frac{\log p}{p^2-p+1} - \frac{8 \log 2}{3} \right) + O \left( \frac{(\log x)^{5/3}}{x} \right). \] Dedekind 函數 $\psi(n)=n \prod_{p\mid n} \left(1+\frac1{p}\right)$ 對應的 Dirichlet 級數是 \[ \sum_{n=1}^{\infty} \frac{\psi(n)}{n^s} = \frac{\zeta(s)\zeta(s-1)}{\zeta(2s)} \quad (\Re s>2), \] 交錯級數對應的 Dirichlet 級數是 \[ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\psi(n)}{n^s} = \frac{2^s-5}{2^s+1} \cdot \frac{\zeta(s)\zeta(s-1)}{\zeta(2s)} \quad (\Re s>2). \] $\psi$ 均值的余項最好的估計也屬于 Walfisz [1, p. 100] \[ \sum_{n\leqslant x} \psi(n) = \frac{15}{2\pi^2} x^2 + O\left( x (\log x)^{2/3} \right). \] 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 研究了某些積性函數倒數的均值, 他們證明了 [5, Corollary 4.2] \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)  + O \left( x^{-1} (\log x)^{2/3} (\log \log x)^{4/3}\right). \end{align*} Bordellès 和 Cloitre [3, Corollary 4, (iii)], László Tóth [4, Theorem 20] 分別研究了交錯級數的情形: \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) + O \left( \frac{(\log x)^2}{x} \right). \end{align*} 除數和函數 $\sigma(n)=\sum_{d\mid n} d$ 的 Dirichlet 級數為 \[ \sum_{n=1}^{\infty} \frac{\sigma(n)}{n^s} = \zeta(s)\zeta(s-1) \quad (\Re s>2), \] 交錯級數的 Dirichlet 級數是 \[\sum_{n=1}^{\infty} (-1)^{n-1} \frac{\sigma(n)}{n^s} = \left(1-\frac{6}{2^s}+\frac{4}{2^{2s}}\right) \zeta(s)\zeta(s-1) \quad (\Re s>2). \] $\sigma$ 均值的余項最佳估計仍屬于 Walfisz [1, p. 99] \[ \sum_{n\leqslant x} \sigma(n) = \frac{\pi^2}{12} x^2 + O\left( x (\log x)^{2/3} \right). \] 作為推論, 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 [5, Corollary 4.1] 中給出了 \[ \sum_{n\leqslant x} \frac1{\sigma(n)} = E \left(\log x + \gamma + F \right) + O\left( x^{-1} (\log x)^{2/3}(\log \log x)^{4/3} \right), \] 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 和 Cloitre [3, Corollary 4, (v)], László Tóth [4, Theorem 23] 分別證明了 \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 \[ K= \sum_{j=0}^{\infty} \frac1{2^{j+1}-1}, \qquad K'= \sum_{j=1}^{\infty} \frac{j}{2^{j+1}-1}. \]

參考文獻

1. A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, 1963.

2. E. Landau, über die Zahlentheoretische Function φ(n) und ihre Beziehung zum Goldbachschen Satz, Nachrichten der Koniglichten Gesellschaft der Wissenschaften zu G?ttingen, Mathematisch Physikalische Klasse, 1900, 177–186.

3. O. Bordellès and B. Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, J. Integer Seq. 16 (2013), Article 13.6.3.

4. László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, J. Integer Seq. 20 (2017), Article 17.2.1.

5. V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Math. J. Okayama Univ. 21 (1979), 155–164.