Intereting Posts

Is local isomorphism totally determined by local rings?
$\mathbb Z_p^*$ is a group.
Understanding the proof that $L_\infty$ norm is equal to $\max\{f(x_i)\}$
If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.
Uniform continuity and translation invariance
Why is $\frac{987654321}{123456789} = 8.0000000729?!$
What's the name of these two surfaces?
Definition of local maxima, local minima
Good Physical Demonstrations of Abstract Mathematics
Induction based on sum of $kth$ powers.
gcd and order of elements of group
Product of spheres embeds in Euclidean space of 1 dimension higher
The matrix specify an algernating $k$-tensor on $V$, and dim$\bigwedge^k(V^*)=1$
Does taking the direct limit of chain complexes commute with taking homology?
What is the importance of Calculus in today's Mathematics?

Prove that if $n \in \mathbb{N}$, then $$\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$$ where $d(n)$ is the divisor function. I have know only information is $d(n)=\sum_{d|n}{1}$.

- $p=4n+3$ never has a Decomposition into $2$ Squares, right?
- Gödel, Escher, Bach: $ b $ is a power of $ 10 $.
- $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?
- Graph and Number theory
- Fermat's 2 Square-Like Results from Minkowski Lattice Proofs
- Quadratic reciprocity via generalized Fibonacci numbers?
- Determine in How many ways $N!$ can be expressed as sum of consecutive numbers
- Prove that $n$ divides $\phi(a^n-1)$, where $\phi$ is Euler's $\phi$-function.
- Every natural number is representable as $\sum_{k=1}^{n} \pm k^5$ … if somebody proves it for 240 integers
- A convergence problem： splitting a double sum

**Outline:** Let $f(n)$ be the function on the left, and $g(n)$ the function on the right.

Both $f$ and $g$ are **multiplicative**.

So we only need to verify that the equality holds when $n$ is a power of a prime.

Multiplicativity then atutomatically gives us the rest.

The number $p^k$ has $k+1$ divisors: in symbols, $d(p^k)=k+1$.

So we want to prove that for any $m$,

$$\sum_0^m (k+1)^3 =\left(\sum_0^m (k+1)\right)^2.$$

Look up the formula for the sum of the first $q$ cubes.

- finding $\int {\tan^{3/2} 3x\sec 3x\,dx}$
- How complicated is the set of tautologies?
- A and B disjoint, A compact, and B closed implies there is positive distance between both sets
- Can we simplify $\int_0^{\pi}\left(\frac{\sin nx}{\sin x}\right)^mdx$?
- Differential Geometry-Wedge product
- Find the range of the given function $f$
- Compute cos(5°) to 5 decimal places with Maclaurin's Series
- Average minimum distance between $n$ points generate i.i.d. with uniform dist.
- Proof negation in Gentzen system
- What are all measurable maps $f:\mathbb C\to\mathbb C$ such that $f(ab)=f(a)f(b)$?
- Generators for $S_n$
- Why is a number field always of the form $\mathbb Q(\alpha)$ for $\alpha$ algebraic?
- How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$
- Convex hexagon $ABCDEF$ following equalities $AD=BC+EF, BE=AF+CD, CF=DE+AB$. Prove that $\frac{AB}{DE}=\frac{CD}{AF}=\frac{EF}{BC}$
- Motivation of the definition of principal series.