@article{dbafc89d15574002b1976dc3848dec0a,

title = "An elementary proof of a theorem of Hardy and Ramanujan",

abstract = "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.",

keywords = "05A17, 11P81, Additive number theory, Asymptotic number theory, Partition function",

author = "{Cohen Antonir}, Asaf and Asaf Shapira",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.",

year = "2024",

month = may,

doi = "10.1007/s11139-023-00808-z",

language = "אנגלית",

volume = "64",

pages = "57--66",

journal = "Ramanujan Journal",

issn = "1382-4090",

publisher = "Springer Netherlands",

number = "1",

}