An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv τ(5m+5) \pmod 5$ (2509.00532v1)

Abstract: We present a proof of Ramanujan's congruences $$p(5n+4) \equiv 0 \pmod 5 \text{ and } \tau(5n+5) \equiv 0 \pmod 5.$$ The proof only requires a limiting case of Jacobi's triple product, a result that Ramanujan knew well, and a technique which Ramanujan used himself to compute values of $\tau(n)$.

• The paper introduces a unified elementary proof that establishes Ramanujan's mod 5 congruences for both partition and tau functions via generating function techniques.
• It leverages Jacobi's triple product identity and logarithmic differentiation to derive a recurrence relation for the coefficients of the partition generating function.
• The proof generalizes to an infinite family of congruences and illustrates the deep interplay between partition theory and modular forms.

An Elementary Proof of Ramanujan's Mod 5 Partition and Tau Congruences

Introduction

This paper presents a unified and elementary proof of two of Ramanujan's most celebrated congruences: $p(5m+4) \equiv 0 \pmod{5}$ for the partition function, and $\tau(5m+5) \equiv 0 \pmod{5}$ for the Ramanujan tau function. The approach leverages a limiting case of Jacobi's triple product identity and a recurrence derived via logarithmic differentiation of generating functions, both techniques familiar to Ramanujan. The proof generalizes to an infinite family of congruences for coefficients of powers of the partition generating function, providing a streamlined and conceptually transparent argument.

Preliminaries and Notation

Let $p(n)$ denote the number of unordered integer partitions of $n$, with generating function

$$P(q) = \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty \frac{1}{1-q^k}.$$

The Ramanujan tau function $\tau(n)$ is defined by

$$\sum_{n=0}^\infty \tau(n+1) q^n = \prod_{k=1}^\infty (1-q^k)^{24}.$$

For a rational $r$, define $P_r(n)$ as the coefficient of $q^n$ in $P(q)^r$:

$P(q)^r = \sum_{n=0}^\infty P_r(n) q^n.$

Thus, $P_1(n) = p(n)$ and $P_{-24}(n) = \tau(n+1)$. The main result is the congruence

$$P_r(5m+4) \equiv 0 \pmod{5} \quad \text{if } r \equiv 1 \pmod{5},$$

which recovers Ramanujan's original congruences as special cases.

Recurrence via Logarithmic Differentiation

A key technical tool is a general recurrence for the coefficients of powers of a generating function, derived by taking logarithmic derivatives. For any power series $P(q)$ and nonzero real $r, s$, the coefficients $P_r(n)$ satisfy

$$\sum_{k=0}^n \left(n - \left(\frac{r}{s} + 1\right)k\right) P_r(n-k) P_s(k) = 0.$$

This is obtained by differentiating $P(q)^r$ and $P(q)^s$, multiplying by $q$, and equating coefficients after eliminating the common logarithmic derivative factor. The recurrence is valid for any $P(q)$ with nonzero constant term, but the focus here is on the partition generating function.

Jacobi's Identity and Its Application

The proof crucially uses Jacobi's identity, a limiting case of the triple product:

$$P(q)^{-3} = \prod_{k=1}^\infty (1-q^k)^3 = \sum_{k=0}^\infty (-1)^k (2k+1) q^{k(k+1)/2}.$$

In terms of $P_{-3}(n)$, this gives

$$P_{-3}(n) = \begin{cases} (-1)^k (2k+1), & n = \frac{k(k+1)}{2}, \ 0, & \text{otherwise}. \end{cases}$$

This explicit formula for $P_{-3}(n)$ allows the recurrence to be written as

$$n P_r(n) = \sum_{j=1}^\infty (-1)^{j+1} (2j+1) \left(n + \left(\frac{r}{3} - 1\right) \frac{j(j+1)}{2}\right) P_r\left(n - \frac{j(j+1)}{2}\right).$$

Proof of the Mod 5 Congruences

The proof proceeds by induction on $m$ for $n = 5m+4$. The base case $m=0$ is checked directly. For the inductive step, the recurrence is analyzed modulo 5. The terms in the sum are grouped according to the residue class of $j$ modulo 5. For $r \equiv 1 \pmod{5}$, all terms in the sum are divisible by 5, either by direct calculation or by the induction hypothesis. This establishes that $P_r(5m+4) \equiv 0 \pmod{5}$ for all $m \geq 0$.

The same method yields companion congruences for other residue classes:

• $P_r(5m+1) \equiv 0 \pmod{5}$ if $r \equiv 0 \pmod{5}$
• $P_r(5m+2) \equiv 0 \pmod{5}$ if $r \equiv 2 \pmod{5}$
• $P_r(5m+3) \equiv 0 \pmod{5}$ if $r \equiv 4 \pmod{5}$

Connections and Implications

The approach highlights the power of generating function manipulations, particularly logarithmic differentiation, in deriving recurrences and congruences for partition-theoretic functions. The recurrence used here is a generalization of classical results, such as Ramanujan's recurrence for $p(n)$ involving the divisor sum $\sigma(n)$, and his recurrence for $\tau(n)$. The method is robust and extends to other moduli and related functions, as explored in the authors' previous work.

The proof also underscores the deep interplay between modular forms, partition theory, and combinatorial generating functions. The explicit use of Jacobi's identity connects the combinatorial structure of partitions to the analytic properties of modular forms.

Future Directions

The techniques in this paper suggest several avenues for further research:

• Extension to congruences modulo other primes, particularly those appearing in Ramanujan's other congruences for $p(n)$ and $\tau(n)$.
• Investigation of similar recurrences and congruences for other families of modular forms and their coefficients.
• Algorithmic applications, such as efficient computation of $p(n)$ and $\tau(n)$ modulo small primes using the derived recurrences.
• Exploration of the combinatorial interpretations of the recurrences and their connections to partition identities.

Conclusion

This paper provides a concise and elementary proof of Ramanujan's mod 5 congruences for the partition and tau functions, unifying them within a broader family of congruences for powers of the partition generating function. The approach, based on generating function recurrences and Jacobi's identity, is both conceptually clear and technically efficient, and it opens the door to further generalizations and applications in partition theory and the theory of modular forms.