# Lecture 0
# Analytic number theory
# Prime numbers le x 
plot([prime_pi(x),x/ln(x)],(1,100))

# Better approximation (prime density)
FL=[sum([factorial(k-1)/(log(x)^k) for k in range(1,n)]) for n in range(2,5)]
plot([prime_pi(x)/x]+FL,(100,1000))

# Partitions, Hardy-Ramanujan
Partitions.options(convention='french',latex='list')
def p(n):
    return Partitions(floor(n)).cardinality()
def hr(n):  
    return exp(pi*sqrt(2*n/3))/(4*n*sqrt(3))

Partitions(5).cardinality()

P5=Partitions(5).list()

for L in P5:
    L, L.ferrers_diagram()

[5]
*****
[4, 1]
*
****
[3, 2]
**
***
[3, 1, 1]
*
*
***
[2, 2, 1]
*
**
**
[2, 1, 1, 1]
*
*
*
**
[1, 1, 1, 1, 1]
*
*
*
*
*

plot([p,hr],(1,30))