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apparently, the Dirichlet hyperbola method is no longer up-to-date, and instead Voronoi's identity is used in order to establish good bounds on the Dirichlet divisor problem.

The same applies to the Piltz divisor problem, and I'm looking for a reference explaining the corresponding generalisation of Voronoi's identity which is as readable as possible - I'm a newcomer to this subject.

I don't need the very sharpest bounds, I just want to understand the modern ("Voronoi") methods that use Bessel functions.

Thanks very much in advance!

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  • $\begingroup$ Most arguments involving a Voronoi-type formulas tend to be somewhat technical. For the original Voronoi formula, you might look at Ivic's book "The Riemann Zeta-Function: Theory and Applications," (I forget the chapter), as well as chapter 4 of Iwaniec and Kowalski's "Analytic Number Theory." The same method yields the Voronoi formula for higher order divisor functions, so I'm not sure where you might find an exposition of the Voronoi formula in that case. $\endgroup$
    – Joshua Stucky
    Commented Jul 30, 2021 at 14:03
  • $\begingroup$ You might want to peruse this arXiv preprint of an article by Soumyarup Banerjee, which has been published in Proc. Amer. Math. Soc. 149 (2021), 1025-1038, @AlgebraicsAnonymous. $\endgroup$
    – Jose Arnaldo Bebita
    Commented Jul 31, 2021 at 1:47
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    $\begingroup$ Thanks @ArnieBebita-Dris, I'd already found that after an on-line search. I'm now reading a book of Titchmarsh and Heath-Brown, which gives a pretty simple explanation, apparently using a different technique due to Landau, which avoids Voronoi's formula, but still gets good results. $\endgroup$
    – Cloudscape
    Commented Jul 31, 2021 at 14:36

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