# Eikonal Blog

## Partition function $p(n)$

Definition: The partition function $p(n):{\Bbb N}\rightarrow{\Bbb N}_0$ is defined by

$p(n) = \hbox{number of partitions of } n$.

Examples: The partitions of 5:

• $5 = 5$,
• $5 = 4+1$,
• $5 = 3+2$,
• $5 = 3+1+1$,
• $5 = 2+2+1$,
• $5 = 2+1+1+1$,
• $5 = 1+1+1+1+1$.

i.e. $p(5)= 7$.

Properties:

• $p(1)=1$, $p(2)=2$, $p(3)=3$, $p(4)=5$, $p(5)=7$, $p(6)=11$, $p(7)=15$, $p(8)=22$, $p(9)=30$, $p(10)=42$, … $p(100)=190,569,292$, … $p(200)=3,972,999,029,388$, … $p(1,000)=24,061,467,864,032,622,473,692,149,727,991$, …
• An asymptotic formula: $p(n) \sim \frac{e^{\pi \sqrt{\frac{2n}3}}}{4n\sqrt{3}}$.
• Generating function: $F(x) :\equiv \prod_{m=1}^\infty \frac1{(1-x^m)} = \sum_{n=0}^\infty p(n) x^n$ (where $|x|<1$).
• $p(n) = \oint_C \frac{dx}{2\pi i} \frac{F(x)}{x^{n+1}}$
• … [the] main contributions [to this integral] come from poles of $F(x)$, [and] these lie at every root of $1$.
• Circle Method: using the function $T_q(n)$ that estimates the contribution [to the above integral] from the poles on the circle ${\cal C}$ that are near the q-th roots of unity. The main formula of the method is: $p(n) = \sum_{q=1}^\infty T_q(n)$. Taking the finite number of terms in that sum gives surprisingly good results, due to the fact that the estimated function has integer values.

Some applications of Circle Method:

• Lagrange’s conjecture: Each natural number can be expressed as the sum of at most four squares.
• Waring’s problem: Every sufficiently large natural number can be expressed as the sum of at most $w(k)$ k-th powers.
• Example: To calculate the exact expression for the number of ways ($R_k(N)$) to write number $N$ as a sum of $k$ squares (i.e. $N = \sum_{i=1}^k n_i^2$) use the $k$-th potency of the function $f(x):\equiv \sum_{n=0}^\infty x^{n^2}$:
• $f(x)^k = \sum_{N=0}^\infty R_k(N) x^N$,
• and, therefore: $R_k(N) = \oint_C \frac{dx}{2\pi i} \frac{f(x)^k}{x^{n+1}}$

Goldbach’s conjecture:

• Every even number greater than two can be represented as a sum of two primes.
• Every odd number greater than five can be represented as a sum of three primes.

## Congruences for $p(n)$

Ramanujan’s congruences:

• $p(5n+4) \equiv 0 (mod \ 5)$,
• $p(7n+5) \equiv 0 (mod \ 7)$,
• $p(11n+6) \equiv 0 (mod \ 11)$.

Congruences found in 1960s:

$p(A n + B) \equiv 0 (mod \ l^k)$

for $l = 13, 17, 19, 23, 29, 31$.

Example: $p(13^2\cdot 97^3 \cdot 103^3 \cdot n - 6,950,975,499,605) \equiv 0 (mod \ 13^2)$.

Theorem (Ono, 2000): For prime $l \ge 5$ there exist infinitely many congruences of the form $p(A n + B) \equiv 0 (mod \ l)$.

Theorem (Ahlgren, 2000): For prime $l \ge 5$ and a positive integer $m$ there exist infinitely many congruences of the form $p(A n + B) \equiv 0 (mod \ l^m)$.

## Modular forms

Definition: Ramanujan’s tau-function:

• $\sum_{n=1}^\infty \tau(n) q^n :\equiv q \prod_{m=1}^\infty (1-q^m)^{24} = q -24 q^2 +252 q^3 - 1,472 q^4 + 4,830 q^5 - - 6,048 q^6 - 16,744 q^7 + 84,480 q^8 - 113,643 q^9 - 115,920 q^{10} + \cdots$

Definition: $\Delta(z) :\equiv \sum_{n=1}^\infty \tau(n) q^n :\equiv q \prod_{m=1}^\infty (1-q^m)^{24}$ for $q :\equiv e^{2\pi i z}$ (with $\Im{z}>0$).

The $\Delta(z)$ has modular symmetries: $\Delta\left(\frac{az+b}{cz+d}\right) = (cz+d)^{12} \Delta(z)$ ($\forall z$) for any integer $M= \begin{pmatrix}a & b \\ c & d\end{pmatrix}$ that is unimodular ($\det M = 1$). In another words, $\Delta(z)$ is a modular form of weight 12.

Definition: The function $f:{\Bbb C}\rightarrow{\Bbb C}$ is a modular form of weight $k$ if it satisfies the modularity equation $f\left(\frac{az+b}{cz+d}\right) = (cz+d)^{k} f(z)$ ($\forall z$, ($\forall M\in{\Bbb Z}^{2\times2}/\{\det M=1\}$)).

The $\Delta(z)$ is a prototype of all modular functions. Some other examples are:

• $\theta(z) :\equiv \sum_{n\in{\Bbb Z}} q^{n^2}$,
• $E(z) :\equiv 1+ 240 \sum_{n=1}^\infty (\sum_{d|n} d^3) q^{n}$.

Some properties of $\tau(n)$:

• $\tau(n) \tau(m) = \tau(nm)$ if $gcd(n,m)=1$.
• Ramanujan’s conjecture: $|\tau(p)| \le 2 p^{11/2}$ for every prime $p$.
• Generalized Ramanujan’s conjecture: if $f(z) = \sum_{n=1}^\infty a(n) q^n$ has weight $k$, then $|a(p)| \le 2 p^{{k-1}/2}$ for all primes $p$.

Congruences for $\tau$ function:

• $\tau(p) \equiv 1+ p^{11} \ (\hbox{mod} \ 691)$ for every prime $p$.
• $\tau(p) \equiv 1+p^{11} \ (\hbox{mod} \ 2^5)$ for $p\ne 2$.
• $\tau(p) \equiv 1+p \ (\hbox{mod} \ 3)$ for $p\ne 3$.
• $\tau(p) \equiv p^{30}+p^{-41} \ (\hbox{mod} \ 5^3)$ for $p\ne 5$.
• $\tau(p) \equiv p+p^4 \ (\hbox{mod} \ 7)$ for $p\ne 7$.
• These are the only congruences of the form $\tau(p) \equiv p^a + p^{11-a} \ (\hbox{mod} \ l^k)$.