The Voronoi Identity via the Laplace Transform

The classical Voronoi identity $$Delta (x) = - frac{2}{pi }sumlimits_{n = 1}^infty {d(n)} left( {frac{x}{n}} right)^{1/2} left( {K_1 (4pi sqrt {xn} ) + frac{pi }{2}Y_1 (4pi sqrt {xn} )} right)$$ is proved in a relatively simple way by the use of the Laplace transform. Here Δ(x) denotes the error term in the Dirichlet divisor problem, d(n) is the number of divisors of n and K_1, Y_1 are the Bessel functions. The method of proof may be used to yield other identities similar to Voronoi's.