2021/02/22 by Hartosh Singh Bal, Bal, Hartosh Singh, Gaurav Bhatnagar +1 · 3 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2102.10804
openalex publication_date 2021/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function σ(n). In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a calculus to associate a generating function with each of these divisor sums. This yields analogues of Ramanujan's recurrence relation for several partition-theoretic functions as well as rk(n) and tk(n), functions counting the number of ways of writing a number as a sum of squares (respectively, triangular) numbers. As by-products of this association, we obtain several convolutions, recurrences and congruences for divisor functions. We give alternate proofs of two classical theorems, one due to Legendre and the other -- Ramanujan's congruence p(5n+4) ≡ 0 \pmod 5.