A path integral approach to lattice point problems. (Q1406919)

Consider the number of lattice points under positive graphs \(y= g(x)\) between \(0\) and \(N+ 1\). Here \(g\in C^2_N\), where \(C^2_N\) denotes the set of all functions \(f\) which are twice continuously differentiable on \([0,N+ 1]\) and where \(f(0)= f(N+ 1)= 0\) and the second derivative \(f''\) is of order \(1/N\). It is shown that the integral of the squared lattice error term over all arcs \(g\) agrees with the conjecture that the error term is of order \(O(N^{1/2+\varepsilon})\). The motivation of proof is given by Feynman's path integral formulation of quantum mechanics.