Omega Results for The Divisor and Circle Problems Using The Resonance Method

We apply the resonance method to obtain large values of general exponential sums with positive coefficients. As applications, we show improved $\mathrm{\Omega }$-bounds for Dirichlet and Piltz divisor problems, Gauss circle Problem, and error term for the mean square of the Riemann zeta function and the Dirichlet $L$-function.