Riemann Zeta Function: 1 + 1/2^s + 1/3^s + 1/4^s + ...
Riemann Hypothesis: All non-trivial zeros of the Riemann Zeta
Function have real part 1/2.

Card trick: Take a stack of poker cards. Slide the topmost card
as far over the edge as it goes without tipping, this is 0.5 
cardlength overhang. Then slide the two topmost cards over as far
as they go without tipping. To find out how much more overhang this
provides, consider the center of mass of the two-card system, which
is halfway along the 1.5 length two-card system, which is 0.75, so
sliding this to the edge gives an extra 0.25, making the total
overhang 0.5+0.25=0.75. Now we can generalize using the formula 
for center of mass. Suppose we are on level N, then we have N-1 cards
with their center of mass at x=0, where the origin is the point of
overhang. But there is an additional card with its center of mass
at x=1/2 in units of cardlength. So x_cm = 1/N * ((N-1)*0 + 1*(1/2))
= 1/(2N). Therefore the maximum total overhang for N cards is
(1/2) Nth partial sum of the harmonic series. Since the harmonic series
diverges, with an arbitrarily large stack of cards, one can achieve 
any desired amount of overhang, so you can have it so the top card is
entirely past the edge of the bottom of the stack.

How do we know the harmonic series diverges? Nicole d'Oresme pointed
out that the elements between 2^n and 2^(n+1) always sum to greater
than one half, and since all the terms are positive they may be rearranged
in this order while preserving the value, and the sum of an infinte
number of terms greater than 1/2 is divergent.

The Basel Problem: Find a closed form for the infinite series of the
inverses of the squares of the positive integers. Euler solved the problem:
the answer is Pi^2/6

Gauss, Dirichlet, Riemann, and Dedekind were all at Gottingen simultaneous
for a while. Gauss' big number theory book was Disquisitiones Arithmeticae

Irish Philosopher George Berkeley said: "And what are these evanescent
increments? They are neither finite quantities, nor quantities infinitely
small, nor yet nothing. May we not call them the ghosts of departed 
quantities?"

Chebyshev Bias - more primes =3(mod 4) than =1(mod 4) 
		also more primes =2(mod 3) than =1(mod 3)
		these are violated, but yet very prominent 

Littlewood Violation - first one is when PI(x) exceeds Li(x)
Littlewood's 1914 Result: Li(x) - PI(x) changes from positive to negative
		and back an infinite number of times - this is connected
		to the Chebyshev Bias
Bays Hudson estimate of first Littlewood Violation: 1.39822*10^316

Van Koch's 1901 Result: If RH is true, then PI(x)=Li(x)+O(sqrt(x)log(x))

Big Oh Definition: Function f is big oh of function g iff for sufficiently
large arguments, the value of f never exceeds some fixed multiple of g.


1/zeta(s) = sum(mu(n)/n^s,1,inf) = probability that s random integers have
	no proper factor in common
The Merten function is defined as M(N)=sum(mu(n),1,N) 
If the Merten function is O(sqrt(x)) then the Riemann Hypothesis is true.

Vis Viva Equation for Astronomy: v = sqrt(M(2/r - 1/a))

Random Matrices: Matrix with elements picked randomly from a Gaussian distribution

Hermitian Matrix: every element is the complex conjugate of the corresponding
	element of its transpose (Note: this implies that the diagonal is real)

Hilbert-Polya Conjecture: The non-trivial zeroes of the Riemann Zeta function
correspond to the eigenvalues of some Hermitian operator.

Riemann, defined a function J. Then he found PI in terms of J, and then
	J in terms of Zeta. This is the connection.

The only well-defined of Hilbert's Problems not solved is the Riemann Hypothesis

Estermann came up with the second proof that sqrt(2) is irrational.

All zeros of the zeta function found so far have irrational complex component