Intereting Posts

If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?
Logical formula of definition of linearly dependent
Is Inner product continuous when one arg is fixed?
Reduced frequency range FFT
Congruence and diagonalizations
Why is the construction of the real numbers important?
Continuity of a convex function
Equivalence of categories involving graded modules and sheaves.
How to express $\cos(20^\circ)$ with radicals of rational numbers?
Locally bounded Family
Show that $n$ lines separate the plane into $\frac{(n^2+n+2)}{2}$ regions
Proving that $n|m\implies f_n|f_m$
Every closed $C^1$ curve in $\mathbb R^3 \setminus \{ 0 \}$ is the boundary of some $C^1$ 2-surface $\Sigma \subset \mathbb R^3 \setminus \{ 0 \}$
Normal subgroups of free groups: finitely generated $\implies$ finite index.
Why study Algebraic Geometry?

The volume of a unit $n$-dimensional ball (in Euclidean space) is

$$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$

The completed Riemann zeta function, or Riemann xi function, is

- Prove: Finitely many Positive Integers $n, s $ such that $n!=2^s(2^{s−2}−1)$
- Bernoulli number type asymptotics
- What is the analytic continuation of the Riemann Zeta Function
- Is there a way to show that $\sqrt{p_{n}} < n$?
- What is the probability that some number of the form $10223\cdot 2^n+1$ is prime?
- A combinatorial sum and identity involving Stirling numbers of the second kind

$$\xi(s) = (s-1) \frac{\frac{s}{2}\Gamma(\frac{s}{2})}{\pi^{s/2}} \zeta(s)$$

Save for the $(s-1)$, the extra factor is exactly the inverse of $V_s$.

Is there any explanation for this, or is it just a funny coincidence?

- Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$
- Contour Integral: $\int^{\infty}_{0}(1+z^n)^{-1}dz$
- Proof $\sum\limits_{k=1}^n \binom{n}{k}(-1)^k \log k = \log \log n + \gamma +\frac{\gamma}{\log n} +O\left(\frac1{\log^2 n}\right)$
- Conjecture about $A(z) = \lim b^{} ( c^{} (z) ) $
- Trying to prove that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$
- Branch cut and $\log(z)$ derivative
- Show that $|z_1 + z_2|^2 < (1+C)|z_1|^2 + \left(1 + \frac{1}{C}\right) |z_2|^2$
- Space of Germs of Holomorphic Function
- an analytic function from unit disk to unit disk with two fixed point
- Where does the theory of Banach space-valued holomorphic functions differ from the classical treatment?

When we prove the functional equation, usually we start by proving the Mellin transform $$\Gamma\left(\frac{s}{2}\right)\pi^{-s/2}\zeta(s)=\int_{0}^{\infty}\psi(x)x^{s/2-1}dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$

where $$\psi(x)=\sum_{n=1}^{\infty}e^{-\pi n^{2}x}.$$

This is where the factor of $\Gamma(s/2)\pi^{-s/2}$ comes from, and this can be proven by writing down the definition of $\Gamma(s/2)$, making a change of variable, and summing over $n$. Instead, when $k$ is an integer we can prove this identity in a different way that makes it clear that this factor of $\Gamma(s/2)\pi^{-s/2}$

is really $A_{k-1}/2$

where $A_{k-1}$

is the surface area of the $k$-dimensional ball.

We have that

$$\int_{-\infty}^{\infty}e^{-\pi n^{2}x^{2}}dx=\frac{1}{n},$$

and so $$\zeta(k)=\sum_{n=1}^{\infty}\frac{1}{n^{k}}=\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}e^{-\pi n^{2}(x_{1}^{2}+\cdots+x_{k}^{2})}dx_{1}\cdots dx_{k}.$$

Switching to spherical coordinates and letting $r^{2}=x_{1}^{2}+\cdots+x_{k}^{2}$, a shell of radius $r$

has size $A_{k-1}r^{k-1}$

and so $$\zeta(k)=A_{k-1}\int_{0}^{\infty}\sum_{n=1}^{\infty}e^{-\pi n^{2}r^{2}}r^{k-1}dr,$$

and then by letting $t=r^{2},$

we then have that $$\frac{2\zeta(k)}{A_{k-1}}=\int_{0}^{\infty}\psi(t)t^{k/2-1}dt.$$

Modifying this proof, one can show directly that $$A_{k-1}=\frac{2\pi^{k/2}}{\Gamma(k/2)}.$$

- How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$
- Finding $\lim_{n\to\infty}M^n$ for the diagonal matrix $M$
- Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
- Finite field, every element is a square implies char equal 2
- How to prove convergence of polynomials in $e$ (Euler's number)
- Lebesgue outer measure of $\cap\mathbb{Q}$
- construct circle tangent to two given circles and a straight line
- What is the fastest way to multiply two digit numbers?
- Subgroups of $S_n$ of index $n$ are isomorphic to $ S_{n-1}$
- A Bound for the Error of the Numerical Approximation of a the Integral of a Continuous Function
- Why must a field with a cyclic group of units be finite?
- Possible order of $ab$ when orders of $a$ and $b$ are known.
- Big-O Interpretation
- What is the proof that the total number of subsets of a set is $2^n$?
- Infinite Series: Fibonacci/ $2^n$