Computing the integer partition function: Addendum

Neil Calkin, Jimena Davis, Kevin James, Elizabeth Perez, Charles Swannack

Date: May 1, 2006

The algorithm of [25] was used with $N_p = 108$ to compute $p(n)$ modulo primes less than $104$ for $n \le 10^9$. Here we list additional statistics to those appearing in [25]. Computations are ongoing. In order to be concise we only list the intermediate results for $10^6,10^7,10^8$ and $10^9$.

In order to examine the conjectures and speculation of [6] we make the following definitions.

Definition 0.1   Let $M \in \mathbb{Z}$ define for any $X \in \mathbb{Z}$

$$\mu(M,X) = \frac{1}{M} \sum_{j=0}^{M-1} \delta_j(M,X) = \frac{1}{M}$$

and

$$\sigma^2(M,X) = \frac{1}{M} \sum_{j=0}^{M-1} \left( \delta_j(M,X) - \frac{1}{M} \right)^2$$

to be the mean and variance of the distribution of $p(n) \pmod{M}$ for $n < X$ respectively.

Definition 0.2   Let $M \in \mathbb{Z}$ and define for any $X \in \mathbb{Z}$

$$\mu_d(M,X) = \frac{1}{M-1} \sum_{j=1}^{M-1} \delta_j(M,X) = \frac{1 - \delta_0(M,X)}{M-1}$$

and

$$\sigma^2_d(M,X) = \frac{1}{M-1} \sum_{j=1}^{M-1} \left( \delta_j(M,X) - \mu_d(M,X) \right)^2$$

to be the mean and variance of the distribution of $p(n) \pmod{M}$ among the non-zero congruence classes mod $M$ for $n < X$.

For each prime $l \ge 5$, define the two integers $\epsilon_l$ and $\delta_l$ to be

$$\epsilon_l = \left( \frac{-6}{l} \right)$$

and

$$\delta_l = \frac{l^2 - 1}{24}.$$

Further, let $S_l$ be the set of $(l+1)/2$ integers

$$S_l = \left\{ \beta \in \{0, 1, \ldots, l-1 \} : \left( \frac{\beta + \delta_l}{l} \right) = 0 \; \textnormal{or} \; - \epsilon_l \right\}.$$

Then, we have the following theorem from [10]

Theorem 0.1   If $l \ge 5$ is prime, $m$ is a positive integer, and $\beta \in S_l$, then there are infinitely many non-nested arithmetic progressions $\{ An + B\} \subset \{ ln + \beta \}$ such that

$$p( A n + B ) \equiv 0 \pmod{l^m}$$

for every integer $n$.

This naturally leads the authors of [10] to the following speculation.

Speculation 0.1   If $l \ge 5$ is prime and $0 \le r < l$, define $\delta'_r(l,X)$ by

$$\delta'_r(l,X) = \frac{\char93 \left\{ n < X : p(n) \equiv r \pm... ...\in S_l \right\} }{ \char93 \left\{ n < X : n \pmod{l} \not\in S_l \right\} }$$

is it true that $\lim_{X \to \infty} \delta'_r(l,X) = \frac{1}{l}$ ?

This leads to the following definition

Definition 0.3   Let $M \in \mathbb{Z}$ define for any $X \in \mathbb{Z}$

$$\mu_p(M,X) = \frac{1}{M} \sum_{j=0}^{M-1} \delta'_j(M,X) = \frac{1}{M}$$

and

$$\sigma_p^2(M,X) = \frac{1}{M} \sum_{j=0}^{M-1} \left( \delta'_j(M,X) - \mu_p(M,X) \right)^2$$

to be the mean and variance of the distribution of $p(n) \pmod{M}$ for $n \not\in S_M$ and $n < X$ respectively.