Best known bounds on the error term for the average order of the sum-of-divisors function

Question

I'm doing some research for a paper I'm writing and I can't seem to find a good non-classical bound for the error term in the following sum (cf. Hardy and Wright): $$\sum_{n \leq x} \sigma_1(n) = \frac{\pi^2}{12} x^2 + O(x \log x).$$ There seem to be many references in a bibliographic search for the corresponding error term in the same sum over the divisor function $d(n) = \sigma_0(n)$, but I'm not having any luck finding a good more modern reference for the case above. I have a friend who claims that you can improve the error term to something like $O(x^{1/3} \log x)$ using modular forms and the hyperbolic method, but these details are still fuzzy to me. Can anyone suggest a reference to the best currently known error terms for this sum? Much appreciated.

Answer 1

The best known result is due to Walfisz (Walfisz, A. -- Weylsche Exponentialsummen in der neueren Zahlentheorie -- 1963 -- approximately on page 100). He proves that

$$\sum_{n \leq X} \sigma_1(n) = \frac{\pi^2}{12}X^2 + O(X (\log X)^{2/3}).$$

The error term cannot possibly be improved past $O(X)$ without introducing a secondary leading term, and it is quite unlikely that one could extract two main terms and an error term of size approximately $O(X^{1/3})$. I would guess that your friend is confused. (If not, then your friend should publish a paper).