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Let$Z$($t$) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term$O$($T$^{1/6}log$T$) is given for the integral of$Z$($t$) over the interval [0,$T$], with special attention paid to the critical cases when the fractional part of $\sqrt{T/2\pi }$ is close to $\frac{1}{4}$ or $\frac{3}{4}$ .

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We quantize a string in the de Sitter background, and we find that the mass spectrum is modified by a term which is quadratic in oscillating numbers, and also proportional to the square of the Hubble constant.

Two overlapping Schwarz algorithms are developed for a discontinuous Galerkin finite element approximation of second order scalar elliptic problems in both two and three dimensions. The discontinuous Galerkin formulation is based on a staggered discretization introduced by Chung and Engquist [<i>SIAM J. Numer. Anal.</i>, 47 (2009), pp. 3820--3848] for the acoustic wave equation. Two types of coarse problems are introduced for the two-level Schwarz algorithms. The first is built on a nonoverlapping subdomain partition, which allows quite general subdomain partitions, and the second on introducing an additional coarse triangulation that can also be quite independent of the fine triangulation. Condition number bounds are established and numerical results are presented.