On divisor sums due to Erdős and Ramanujan

Abstract

Let d(n) denote the number of divisors of a positive integer n. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum ∑_{n ≤ x} 1/d(n), with Ramanujan having introduced an asymptotic formula for this sum with an explicit evaluation for the constant A₁ for the leading term A₁ x/(√(log x)). Gabdullin et al. recently considered a hybrid of this problem and the Titchmarsh divisor problem concerning ∑_{p≤ x} d(p−1), proving that ∑_{p≤ x} 1/(d(p−1)) asymp x/((log x)^3/2). This result and Erdős's asymptotic formula $ \sum_{n \leq x} d(d(n)) \sim c_1 \, x \log \log x $ for a constant c₁ ∈ (0, ∞) lead us to consider the hybrid ∑_{n ≤ x} 1/d(d(n)) of the Erdős and Ramanujan divisor sums. In this paper, we prove that there exists a constant c₃ ∈ (0, ∞) such that ∑_{n ≤ x} 1/d(d(n)) ∼ c₃ x/( log log x), through a combined application of Golomb's estimate for powerful numbers, Turán's quantitative form of the Hardy-Ramanujan theorem, and a fourth-moment estimate for the number of distinct prime divisors due to Granville and Soundararajan. Moreover, an explicit formula for the constant c₃ is given.

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BibTeX RIS

BibTeX

@misc{campbell2026divisor,
  title = {{On divisor sums due to Erdős and Ramanujan}},
  author = {John M. Campbell},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.00695},
  eprint = {2605.00695},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.00695},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On divisor sums due to Erdős and Ramanujan
AU  - John M. Campbell
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2605.00695
UR  - https://arxiv.org/abs/2605.00695
AB  - Let d(n) denote the number of divisors of a positive integer n. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum ∑_n ≤ x 1/d(n), with Ramanujan having introduced an asymptotic formula for this sum with an explicit evaluation for the constant A_1 for the leading term A_1 x/(√(log x)). Gabdullin et al. recently considered a hybrid of this problem and the Titchmarsh divisor problem concerning ∑_p≤ x d(p−1), proving that ∑_p≤ x 1/(d(p−1)) asymp x/((log x)^3/2). This result and Erdős's asymptotic formula $ \sum_{n \leq x} d(d(n)) \sim c_1 \, x \log \log x $ for a constant c_1 ∈ (0, ∞) lead us to consider the hybrid ∑_n ≤ x 1/d(d(n)) of the Erdős and Ramanujan divisor sums. In this paper, we prove that there exists a constant c_3 ∈ (0, ∞) such that ∑_n ≤ x 1/d(d(n)) ∼ c_3 x/( log log x), through a combined application of Golomb's estimate for powerful numbers, Turán's quantitative form of the Hardy-Ramanujan theorem, and a fourth-moment estimate for the number of distinct prime divisors due to Granville and Soundararajan. Moreover, an explicit formula for the constant c_3 is given.
ER  -

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