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The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d – 1 is a prime power. In this paper, we consider the case when d – 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost” Ramanujan graphs for almost every value of d. More precisely, for any fixed ℇ > 0 and for almost every value of d (in the sense of natural density), there are infinitely many d-regular graphs such that all the non-trivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ℇ) . 2000 Mathematics Subject Classification: 05C50, 15A18, 11N05, 11F30. [ABSTRACT FROM AUTHOR]

For a fixed hyperelliptic curve C given by the equation y2 = f( x) with f ∈ ℤ[ x] having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm whose equation is given by my2 = f( x). More precisely, we study the Dirichlet series where the summation is over all non-zero squarefree integers. We show that converges for ℜ( s) > 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in ℤ[ x]. [ABSTRACT FROM AUTHOR]

If L ⊆ ℤ n is a lattice of finite index, let , where the sum is over all lattices L′ with L ⊆ L′ ⊆ ℤ n. Using the automorphic theory of GL2 n, a generalization of the Voronoi-Oppenheim summation formula is proved for these generalized divisor functions. Bessel distributions and Hankel transforms associated with the Shalika model for the degenerate principal series representations of GL2 n(ℝ) are principal tools. [ABSTRACT FROM AUTHOR]

On a smooth compact manifold M, the cohomology of the complex of differential forms is isomorphic to the ordinary cohomology by the classical theorem of de Rham. When M has a Riemannian metric g, the harmonic forms constitute a subcomplex of the de Rham complex because the Laplacian commutes with exterior differentiation. When ( M, g) has no boundary, all of its harmonic forms are closed, and hence the cohomology of this subcomplex is isomorphic to the ordinary cohomology by the classical theorem of Hodge. But when the boundary of ( M, g) is non-empty, it is possible for a p-form to be harmonic without being closed, and some of these, which are exact, although not the exterior derivatives of harmonic p – 1-forms, represent an “echo” of the ordinary p – 1-dimensional cohomology within the p-dimensional harmonic cohomology. [ABSTRACT FROM AUTHOR]

We establish estimates for certain families of character sums over finite fields, including those which arise in Lillian Pierce's recent work [P] on estimating the 3-part of the class number of quadratic fields. [ABSTRACT FROM AUTHOR]

We prove a nontrivial bound of O(| D|27/56+ε) for the 3-part of the class number of a quadratic field ℚ(√ D) by using a variant of the square sieve and the q-analogue of van der Corput's method to count the number of squares of the form 4 x3 − dz2 for a square-free positive integer d and bounded x, z. [ABSTRACT FROM AUTHOR]

We determine all integers as sums of three integral squares over all imaginary quadratic fields. [ABSTRACT FROM AUTHOR]