解析数论笔记（1）

the Möbius funtion

\[\begin{align*} \mu(n)=\begin{cases} 1 \,\,\,\,\,\,\text{ if } n=1 \\ (-1)^k \,\,\,\,\,\,\text{ if } a_1=a_2=a_3=...=a_k=1\\ 0 \,\,\,\,\,\,\text{ otherwise } \end{cases} \end{align*}\]

the Euler totient function

\[\begin{align*} \varphi(n)=\sum_{ \substack{1\le k\lt n \\(k,n)=1} }1 \end{align*}\]

the Mangoldt function

\[\begin{align*} \Lambda(n)=\begin{cases} \log(p) \,\,\,\,\,\,\text{ if } n=p^m \\ 0 \,\,\,\,\,\,\text{ otherwise } \end{cases} \end{align*}\]

Liouville’s function

\[\begin{align*} \lambda(n)=\begin{cases} 1 \,\,\,\,\,\,\text{ if } n=1 \\ (-1)^{a_1+a_2+a_3+...+a_k} \,\,\,\,\,\,\text{ if } n=p_1^{a_1}p_2^{a_2}...p_k^{a_k} \end{cases} \end{align*}\]

the divisor functions

\[\begin{align*} \sigma_\alpha(n)=\sum_{ d|n}d^\alpha \end{align*}\] \[\begin{align*} d(n)=\sum_{ d|n}1 \end{align*}\]

the identity function

\[\begin{align*} I(n)=\left[\frac1n\right] \end{align*}\]

the Riemann zeta function

\[\begin{align*} \zeta(n)=\sum_{ n=1}^\infty \frac{1}{n^s} \end{align*}\]

the number of primes

\[\begin{align*} \pi(x)=\sum_{ 2\le p\le x}1 \end{align*}\]

Chebyshev’s $\psi$-function

\[\begin{align*} \psi(n)=\sum_{ n\le x}\Lambda (n) \end{align*}\]

Chebyshev’s $\vartheta$-function

\[\begin{align*} \vartheta(n)=\sum_{ p\le x}\log (p) \end{align*}\]

partial sums of the Möbius funtion

\[\begin{align*} M(x)=\sum_{ n\le x}\mu(n) \end{align*}\]

Dirichlet characters

\[\begin{align*} \chi=\chi_f(n)=\begin{cases} f(\hat n) \,\,\,\,\,\,\text{ if } (n,k)=1 \\ 0 \,\,\,\,\,\,\text{ if } (n,k)\gt 1 \end{cases} \end{align*}\]

the Dirichlet-L functions

\[\begin{align*} L(s,\chi)=\sum_{ n=1}^\infty \frac{\chi(n)}{n^s} \end{align*}\]

Ramanujan’s sum

\[\begin{align*} c_k(n)=\sum_{ \substack{m \mod k\\ (m,k)=1}} e^{2\pi i mn/k} \end{align*}\]

the Gauss sum associated with $\chi$

\[\begin{align*} G(n,\chi)=\sum_{ m=1}^k \chi(m)e^{2\pi i mn/k} \end{align*}\]

$\nu$

\[\begin{align*} \nu(n)=\begin{cases} 0 \,\,\,\,\,\,\text{ if } n=1 \\ k \,\,\,\,\,\,\text{ if } n=p_1^{a_1}p_2^{a_2}...p_k^{a_k} \end{cases} \end{align*}\]

$\kappa$

\[\begin{align*} \kappa(n)=\begin{cases} 1 \,\,\,\,\,\,\text{ if } n=1 \\ a_1a_2...a_k \,\,\,\,\,\,\text{ if } n=p_1^{a_1}p_2^{a_2}...p_k^{a_k} \end{cases} \end{align*}\]

the hurwitz zeta function

\[\begin{align*} \zeta(s,a)=\sum_{ n=0}^\infty \frac{1}{(n+a)^s} \end{align*}\]

the periodic zeta function

\[\begin{align*} F(x,s)=\sum_{ n=0}^\infty \frac{e^{2\pi i n x}}{n^s} \end{align*}\]

$\psi_1$

\[\begin{align*} \psi_1(x)=\int_{ 1}^\infty \psi(t)dt \end{align*}\]

Riemann’s prime counting function

\[\displaystyle J \left( x \right) = \sum _{ 1 < p ^{ k } \leqslant x } \frac{ 1 }{ k }\]