Intereting Posts

Showing $(P\to Q)\land (Q\to R)\equiv (P\to R)\land$ {without truth table}
Integral inequality (Cauchy-Schwarz)
Cauchy Problem – Real Analysis
Remark 4.31 in Baby Rudin: How to verify these points?
Does the sequence $\{\sin^n(n)\}$ converge?
Evaluating: $\int \frac{t}{\cos{t}} dt$
Nuking the Mosquito — ridiculously complicated ways to achieve very simple results
How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?
Confused with proof that all Cauchy sequences of real numbers converge.
Convergence in quadratic mean and in mean
Limit of positive sequence $(f_n)$ defined by $f_n(x)^2=\int^x_0 f_{n-1}(t)\mathrm{d}t$
What are some classic fallacious proofs?
Showing a metric space is bounded.
help in understanding tangent vectors
What is known about the minimal number $f(n)$ of geometric progressions needed to cover $\{1,2,\ldots,n\}$, as a function of $n$?

This problem appears as a (starred!) exercise in D. Zagier’s notes on modular forms. I have to admit that I have no idea how to do it.

Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual.

This identity is traditionally obtained by using the fact that the space $M_8(\text{SL}_2(\mathbf Z))$ of modular forms of weight $8$ and level $1$ is $1$-dimensional, and contains both $E_4^2$ and $E_8$ ($E_k$ the Eisenstein series of weight $k$). Using this, it’s a piece of cake (simply a matter of comparing coefficients).

- What is wrong with this proof that there are no odd perfect numbers?
- Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality?
- Is $p(p + 1)$ always a friendly number for $p$ a prime number?
- Proof regarding Robin's inequality (RI).
- When is the sum of divisors a perfect square?
- For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$

Without using modular forms, though, I am stumped.

- a question about permutation in the digits in the decimal system
- Use Fermat's little theorem to find remainder of powers
- Value of cyclotomic polynomial evaluated at 1
- For what $n$ and $m$ is this number a perfect square?
- Show $\,1897\mid 2903^n - 803^n - 464^n + 261^n\,$ by induction
- If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
- If a and b are relatively prime and ab is a square, then a and b are squares.
- Prove that $\phi(n) \geq \sqrt{n}/2$
- Find when three numbers have the same remainder when divided by the same number
- If $\gcd(a,b)=1$, is $\gcd(a^x-b^x,a^y-b^y)=a^{\gcd(x,y)}-b^{\gcd(x,y)}$?

- isoperimetric inequality using Fourier analysis
- Solving an integral coming from Perron's formula
- Regular monomorphisms of commutative rings
- Pre-requisites needed for algebraic number theory
- A question about Hartshorne III 12.2
- Points and lines covering them
- Orthogonality and linear independence
- Partial derivative of integral: Leibniz rule?
- What is the value of $\prod_{i=1}^\infty 1-\frac{1}{2^i}$?
- Questions around the number of subgroups of a $p$-group
- How to find center of a conic section from the equation?
- Calculate: $\lim\limits_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$
- Linear optimization problem.
- Predicting Real Numbers
- A binary operation, closed over the reals, that is associative, but not commutative