Update mobius.md

diff --git a/docs/math/mobius.md b/docs/math/mobius.md
index e6f636e..8007926 100644 (file)
--- a/docs/math/mobius.md
+++ b/docs/math/mobius.md
@@ -800,19 +800,19 @@ $$
 $$
 \begin{aligned}
 & \sum_{i=1}^n\sum_{j=1}^ni\cdot j\cdot \gcd(i,j)\\
-&=\sum_{i=1}^n\sum_{j=1}^ni\cdot j
+=&\sum_{i=1}^n\sum_{j=1}^ni\cdot j
 \sum_{d \mid i,d \mid j}\varphi(d)\\
-&=\sum_{d=1}^n\sum_{i=1}^n
+=&\sum_{d=1}^n\sum_{i=1}^n
 \sum_{j=1}^n[d \mid i,d \mid j]\cdot i\cdot j
 \cdot\varphi(d)\\
-&=\sum_{d=1}^n
+=&\sum_{d=1}^n
 \sum_{i=1}^{\left\lfloor\frac{n}{d}\right\rfloor}
 \sum_{j=1}^{\left\lfloor\frac{n}{d}\right\rfloor}
 d^2\cdot i\cdot j\cdot\varphi(d)\\
-&=\sum_{d=1}^nd^2\cdot\varphi(d)
+=&\sum_{d=1}^nd^2\cdot\varphi(d)
 \sum_{i=1}^{\left\lfloor\frac{n}{d}\right\rfloor}i
 \sum_{j=1}^{\left\lfloor\frac{n}{d}\right\rfloor}j\\
-&=\sum_{d=1}^nF^2\left(\left\lfloor\frac{n}{d}\right\rfloor\right)\cdot d^2\varphi(d)
+=&\sum_{d=1}^nF^2\left(\left\lfloor\frac{n}{d}\right\rfloor\right)\cdot d^2\varphi(d)
 \left(F(n)=\frac{1}{2}n\left(n+1\right)\right)\\
 \end{aligned}
 $$
@@ -909,10 +909,10 @@ $$
 $$
 \begin{aligned}
 &\sum_{d=1}^{n}d^3\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{d}\rfloor}ij\cdot[\gcd(i,j)=1]\\
-&=\sum_{d=1}^{n}d^3\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{d}\rfloor}ij\sum_{t\mid \gcd(i,j)}\mu(t)\\
-&=\sum_{d=1}^{n}d^3\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{d}\rfloor}ij\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}\mu(t)[t\mid \gcd(i,j)]\\
-&=\sum_{d=1}^{n}d^3\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}t^2 \mu(t)\sum_{i=1}^{\lfloor\frac{n}{td}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{td}\rfloor}ij[1\mid \gcd(i,j)]\\
-&=\sum_{d=1}^{n}d^3\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}t^2 \mu(t)\sum_{i=1}^{\lfloor\frac{n}{td}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{td}\rfloor}ij
+=&\sum_{d=1}^{n}d^3\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{d}\rfloor}ij\sum_{t\mid \gcd(i,j)}\mu(t)\\
+=&\sum_{d=1}^{n}d^3\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{d}\rfloor}ij\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}\mu(t)[t\mid \gcd(i,j)]\\
+=&\sum_{d=1}^{n}d^3\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}t^2 \mu(t)\sum_{i=1}^{\lfloor\frac{n}{td}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{td}\rfloor}ij[1\mid \gcd(i,j)]\\
+=&\sum_{d=1}^{n}d^3\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}t^2 \mu(t)\sum_{i=1}^{\lfloor\frac{n}{td}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{td}\rfloor}ij
 \end{aligned}
 $$
 
@@ -927,9 +927,9 @@ $$
 $$
 \begin{aligned}
 &\sum_{d=1}^{n}d^3\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}t^2 \mu(t)\sum_{i=1}^{\lfloor\frac{n}{td}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{td}\rfloor}ij\\
-&=\sum_{d=1}^{n}d^3\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}t^2 \mu(t)\cdot F^2\left(\left\lfloor\frac{n}{td}\right\rfloor\right)\\
-&=\sum_{T=1}^{n}F^2\left(\left\lfloor\frac{n}{T}\right\rfloor\right) \sum_{d\mid T}d^3\left(\frac{T}{d}\right)^2\mu\left(\frac{T}{d}\right)\\
-&=\sum_{T=1}^{n}F^2\left(\left\lfloor\frac{n}{T}\right\rfloor\right) T^2\sum_{d\mid T}d\cdot\mu\left(\dfrac{T}{d}\right)
+=&\sum_{d=1}^{n}d^3\sum_{t=1}^{\lfloor\frac{n}{d}\rfloor}t^2 \mu(t)\cdot F^2\left(\left\lfloor\frac{n}{td}\right\rfloor\right)\\
+=&\sum_{T=1}^{n}F^2\left(\left\lfloor\frac{n}{T}\right\rfloor\right) \sum_{d\mid T}d^3\left(\frac{T}{d}\right)^2\mu\left(\frac{T}{d}\right)\\
+=&\sum_{T=1}^{n}F^2\left(\left\lfloor\frac{n}{T}\right\rfloor\right) T^2\sum_{d\mid T}d\cdot\mu\left(\dfrac{T}{d}\right)
 \end{aligned}
 $$
 
@@ -954,7 +954,7 @@ $$
 
 $$
 \begin{aligned}
-&g(n)=\sum_{i=1}^n\mu(i)t(i)f\left(\left\lfloor\frac{n}{i}\right\rfloor\right)\\
+g(n)&=\sum_{i=1}^n\mu(i)t(i)f\left(\left\lfloor\frac{n}{i}\right\rfloor\right)\\
 &=\sum_{i=1}^n\mu(i)t(i)
 \sum_{j=1}^{\left\lfloor\frac{n}{i}\right\rfloor}t(j)
 g\left(\left\lfloor\frac{\left\lfloor\frac{n}{i}\right\rfloor}{j}\right\rfloor\right)\\