Eikonal Blog

2010.11.02

Readings in number theory

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).

Sources


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