An elementary proof of a theorem of Hardy and Ramanujan

Asaf Cohen Antonir, Asaf Shapira

Research output: Contribution to journalArticlepeer-review


Let Q(n) denote the number of integers 1≤q≤n whose prime factorization q=∏i=1tpiai satisfies a1≥a2≥⋯≥at. Hardy and Ramanujan proved that (Formula presented.) Before proving the above precise asymptotic formula, they studied in great detail what can be obtained concerning Q(n) using purely elementary methods, and were only able to obtain much cruder lower and upper bounds using such methods. In this paper, we show that it is in fact possible to obtain a purely elementary (and much shorter) proof of the Hardy–Ramanujan Theorem. Towards this goal, we first give a simple combinatorial argument, showing that Q(n) satisfies a (pseudo) recurrence relation. This enables us to replace almost all the hard analytic part of the original proof with a short inductive argument.

Original languageEnglish
Pages (from-to)57-66
Number of pages10
JournalRamanujan Journal
Issue number1
StatePublished - May 2024


  • 05A17
  • 11P81
  • Additive number theory
  • Asymptotic number theory
  • Partition function

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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