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I'm working on a problem right now in which I need an upper bound for an exponential sum of the form $$ \tag{1} \sum_{N < n \leq 2N} \tau_3(n) e(f(n)), $$ where $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ and $f$ is some function whose derivatives I understand. Jutila has a result (Theorem 4.6 of his Lectures on a Method in the Theory of Exponential Sums) which gives an estimate for the sum $$ \tag{2} \sum_{N < n \leq 2N} \tau_2(n) e(f(n)). $$ Forgive me for not just quoting the result, as it is somewhat lengthy to state. The proof of the estimate for (2) relies on the Voronoi summation formula for $$ \sum_{n\leq x} \tau_2(n) e\left(\frac{an}{q}\right), $$ of which there exist analogues for $\tau_3$ (and all the higher order divisor functions as well), for instance, from the work of Ivic. The application of the summation formula is the only portion of the proof that would be substantially different. Certainly an estimate of the form given in Theorem 4.6 of Jutila's notes can be obtained for (2), as I'm confident I could sit down and write out the details myself. However, before I invest the time and effort to do so, I wanted to see if anyone knows of any papers that have already done this.

I should note that writing $\tau_3(n) = \sum_{d\mid n} \tau_2(d)$ and attempting to estimate (1) by switching divisors and applying (2) is not sufficient for the problem I'm working on, hence my question.

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  • $\begingroup$ Not directly helpful, but your problem is at least superficially similar to t-aspect subconvexity for GL(3) L-functions. I'm not sure what the state of the art is, but maybe this paper of Munshi gives a starting point: arxiv.org/abs/1301.1007 $\endgroup$
    – A of E
    Commented Jul 8, 2021 at 6:41
  • $\begingroup$ @AofE Thanks for the comment. It's funny you link this paper; I actually just went through it in a topics class in the spring. I'll have to take a second look at it. $\endgroup$
    – Joshua Stucky
    Commented Jul 8, 2021 at 7:46
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    $\begingroup$ If you sit down and do this you will find the issue is when Jutila applies large sieve. Instead of a spacing problem with Kloosterman fractions you will need to estimate correlations between sums of Kloosterman sums. $\endgroup$
    – ZzzzzzzzzzZzzzzzzzZZZZZZZXZZ
    Commented Aug 24, 2021 at 12:03

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