Let \(d(x)\) denote the number of positive divisors of \(x\) if \(x\) is a natural number, and \(d(x)=0\) otherwise. The error term \(\Delta(x)\) in the Dirichlet divisor problem may be defined, for \(x>0\), by \[ \sum_{n<x} d(n)+ {\textstyle {1\over 2}}d(x)= x\log x+ (2\gamma-1)x+ {\textstyle {1\over 4}}+ \Delta(x). \] The aim of the author is to give a relatively simple new proof of Voronoi’s famous identity for \(\Delta(x)\) in the form \[ \Delta(x)= {2\over\pi} \sum_{n=1}^ \infty\;{{d(n)}\over n} \int_ 0^ \infty \cos(2\pi u) \sin\biggl( {{2\pi nx} \over u} \biggr) du.\tag{1} \] The series is boundedly convergent in any interval \([x_ 1,x_ 2]\subset (0,\infty)\), and uniformly convergent in any such interval free from integers. The method of proof, which is susceptible to generalizations, is to deal with \[ \Delta(x,v):= {2\over\pi} \sum_{n=1}^ \infty {{d(n)} \over n} F(nx,v), \qquad F(t,v):= \int_ 0^ \infty e^{-v(u+t/u)} \cos(2\pi u) \sin\biggl( {{2\pi t} \over u} \biggr) du \] for \(0\leq v\leq 1/2\), using \(\lim_{v\to 0+} \Delta(x,v)= \Delta(x)\). Usually one expresses the integral in (1) in terms of the Bessel functions as \[ -\sqrt{nx} \biggl(K_ 1 (4\pi\sqrt{nx}) + {\pi\over 2} Y_ 1 (4\pi \sqrt{nx}) \biggr), \] but it is shown how the integral in (1) can be evaluated asymptotically without recourse to the theory of Bessel functions. This is another nice aspect of the author’s approach to the Voronoi formula.
For the entire collection see [Zbl 0772.00022].