This paper estimates the upper bound of the minimum prime quadratic residue module a prime and gives the asymptotic formula of &∑_{p≤x;f(p)=r} 1& (r is a prime).

In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume Vg,n of the moduli spaces of hyperbolic surfaces of genus g with n punctures as g → ∞. Then we discuss basic geometric properties of a random hyperbolic surface of genus g with respect to the Weil-Petersson measure as g → ∞.

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We establish sharp asymptotics for the$L$^{$p$}-norm of Hermite polynomials and prove convergence in distribution of suitably normalized Wick powers. The results are combined with numerical integration to study an extremal problem on Wiener chaos.

We generalize the results of Adams–Schoenfeld [2], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each covering a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic Seifert surfaces by giving bounds on the width invariant in the presence of such a surface. Finally, we utilize these examples to demonstrate that the Six Theorem is sharp for knot complements in the 3-sphere.