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The mean-square value of the divisor function
Published online by Cambridge University Press: 14 June 2016
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The behaviour of the divisor function d (n) is rather tricky. For a prime p, we have d(p) = 2, but if n is the product of the first k primes then, by Chebyshev's estimate for the prime counting function [1, Theorem 414], we have 
so that
for such n then, d (n) is ‘unusually large’ — it can exceed any fixed power of log n, for example.
In [2] Jameson gives, amongst other things, a derivation of Dirichlet's theorem, which shows that the mean-value of the divisor function in an interval containing n is log n. However, the result is somewhat deceptive because, for most n, the value of d (n) is substantially smaller than log n.
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References
1.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (4th edn.), Oxford (1960).Google Scholar
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4.Wilson, B. M., Proofs of some formulae enunciated by Ramanujan, Proc. London Math. Soc. (2) 21 (1922) pp. 235–255.Google Scholar
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