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This is from Hardy and Wright’s An Introduction to the Theory of Numbers, Section 18.2. "The average order of $d(n)$". Here, $d(n)$ denotes the number of divisors of $n$. The section states that using some theorems of Ramanujan, the sum $$d^2(1)+...+d^2(n)$$ is of order $n(\log n)^{2^2-1}=n(\log n)^3$ and the sum $$d^3(1)+...+d^3(n)$$ is of order $n(\log n)^{2^3-1}=n(\log n)^7$, and so on.

It's not clear to me exactly what "is of order" means. What exactly are the theorems they are referring to?

I'm wondering more specifically if given a positive integer $k$, there is a constant $a_k$ such that $$\frac{d^k(1)+...+d^k(n)}{n(\log n)^{2^k-1}}$$ converges to $a_k$ as $n$ goes to infinity. If so, what is the constant $a_k$? I think the answer from @marko-reidel to this question variance of number of divisors has the answer for $k=2$, and I can somewhat follow the steps for that case. But, I do not see how to generalize it for $k>2$. Does anyone know what Ramanujan's theorems say about this or are there any publications that would have the answer? If it can be explained in an undergraduate level, that would be great.

suckling pig
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John L
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    You can safely write $\log^kn$ for $(\log n)^k.$ – Thomas Andrews May 16 '25 at 15:28
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    in the notes at the end of the chapter, they refer to Ramanujan Collected Papers pages 133-135, which is a brief article from 1916. They mention B. M. Wilson (1922) in the Proceedings but no title. – Will Jagy May 16 '25 at 17:03

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In other parts of the book, Hardy and Wright use $f$ is of order $g$ for the weaker relation $A g < f < B g$ for some constants $A$ and $B$.

So the limit you're considering may be doesn't exists, or may be does exists but does not follows from Ramanujan's theorems.

jjagmath
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the original, and the correct title for the Wilson reference, which gives proofs. See also https://en.wikipedia.org/wiki/Bertram_Martin_Wilson

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Will Jagy
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  • If g is an integer, is the left hand side of equation (9) asymptotically equal to the first term on the right hand side (that is the ratio converges to 1)? The $A_1$ in equation (9) is not the same as the $A_1$ defined after equation (7) and I assume it depends on g, right? Is there an explicit formula for $A_1$? – John L May 28 '25 at 14:05