Zeros of the Epstein zeta function to the right of the critical line

Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field ${{\mathbb{Q}}(\sqrt D)}$ . We denote by N E (σ1, σ2, T) the number of zeros of E(s, Q) in the rectangle σ1 < Re(s) ≤ σ2 and T ≤ Im (s) ≤ 2T, where 1/2 < σ1 < σ2 < 1 are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for N E (σ1, σ2, T), obtaining a saving of a power of log T in the error term.

Equivalence of zeta function technique and Abel–Plana formula in regularizing the Casimir energy of hyper-rectangular cavities

Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation. By contrast, Abel–Plana formula method permits explicit separation of divergent terms. In regularizing the Casimir energy for a massless scalar field in a D-dimensional rectangular box, we give the rigorous proof of the equivalence of the two methods by deriving the reflection formula of Epstein zeta function from repeatedly application of Abel–Plana formula and giving the physical interpretation of the infinite integrals. Our study may help with the confidence of choosing any regularization method at convenience among the frequently used ones, especially the zeta function method, without the doubts of physical meanings or mathematical consistency.