Double Exponential method for Riemann Zeta, Lerch and Dirichlet L-functions

Abstract: We provide efficient methods to evaluate the Riemann zeta, the Lerch zeta and the Dirichlet $L$-functions. The method uses the Riemann-Siegel (RS) type formulas and a modified double exponential (MDE) quadrature method near the saddle point of appropriate integrands. We provide simplified derivations of the RS formulas, containing finite series sums and residual integrals, for the Lerch and the Dirichlet $L$-functions. The MDE method allows us to remove the contribution of the singularities near the contour of integration for the residual integrals. The method allows us to evaluate the residual integrals to any prescribed accuracy. The numerical cost of evaluating them is minimal compared to the main series sums. Thus even highly oscillatory integrands for these functions can be evaluated with the same complexity as the RS formula. In particular, the numerical complexity to evaluate the $\zeta(s)$ and the Lerch zeta function with $s=\sigma+i t$ scales as $\sqrt{t}$, and for Dirichlet $L$-function it scales as $\max(q, \sqrt{q t})$. The method allows the automatic setting of integral cutoffs and finite discretization size to achieve prescribed accuracy. Furthermore, it ensures that every halving of the discretization interval almost leads to doubling the accuracy of the results. Thus, it allows for a further check on the accuracy of the results obtained.