几个积性函数的均值

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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

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 Göttingen, 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.

几个积性函数的均值

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原文地址：https://www.cnblogs.com/pengdaoyi/p/10503900.html