We explore the effect of zeros at the central point on nearby zeros of elliptic-curve $L$-functions, especially for one-parameter families of rank $r$ over $\Q$. By the Birch and Swinnerton-Dyer conjecture and Silverman's specialization theorem, for $t$ sufficiently large the $L$-function of each curve $E_t$ in the family has $r$ zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with $r$. There is greater repulsion in the subset of curves of rank $r+2$ than in the subset of curves of rank $r$ in a rank-$r$ family. For curves with comparable conductors, the behavior of rank-$2$ curves in a rank-$0$ one-parameter family over $\Q$ is statistically different from that of rank-$2$ curves from a rank-$2$ family. In contrast to excess-rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the $r$ family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank, and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank $0$, $2$, or $4$ and comparable conductors from one-parameter families of rank $0$ or $2$ over $\Q$.

We document the discovery of two generating functions for $\zeta(2n+2)$, analogous to earlier work for $\zeta(2n+1)$ and $\zeta(4n+3)$, initiated by Koecher and pursued further by Borwein, Bradley, and others.

The geometric structure on a closed orientable hyperbolic 3-manifold determines a discrete faithful representation $\rho$ of its fundamental group into $\mathrm{SO^{+}(3,1)}$, unique up to conjugacy. Although Mostow rigidity prohibits us from deforming $\rho$, we can try to deform the composition of $\rho$ with inclusion of $\mathrm{SO^{+}(3,1)}$ into a larger group. In this sense, we have found by exact computation a small number of closed manifolds in the Hodgson-Weeks census for which $\rho$ deforms into $\mathrm{SL(4,\mathbb R)}$, thus showing that the hyperbolic structure can be deformed in these cases to a real projective structure. In this paper we describe the method for computing these deformations, particular attention being given to the manifold Vol3.

Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a hundred-year-old lower bound on the maximal packing density due to Minkowski in $d$-dimensional Euclidean space $\mathbb{R}^d$. The asymptotic behavior of this bound is controlled by $2^{-d}$ in high dimensions. Using an optimization procedure that we introduced earlier and a conjecture concerning the existence of disordered sphere packings in $\mathbb{R}^d$, we obtain a conjectural lower bound on the density whose asymptotic behavior is controlled by $2^{-0.77865\ldots \,d}$, thus providing the putative exponential improvement of Minkowski's bound. The conjecture states that a hard-core nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in $\mathbb{R}^d$ for asymptotically large $d$ if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical packing constructions in low dimensions, known necessary conditions that have relevance only in very low dimensions, and the fact that we can recover the forms of known rigorous lower bounds. A byproduct of our approach is an asymptotic conjectural lower bound on the average kissing number whose behavior is controlled by $2^{0.22134\ldots \,d}$, which is to be compared to the best known asymptotic lower bound on the individual kissing number of $2^{0.2075\ldots \,d}$. Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies to obtain upper bounds on the density, and hence has implications for linear programming bounds. This connection proves that our density estimate can never exceed the Cohn--Elkies upper bound, regardless of the validity of our conjecture.

We present a numerical study for the first Dirichlet eigenvalue of certain classes of planar regions. Based on this, we propose new types of bounds and establish several conjectures regarding the dependence of this eigenvalue on the perimeter and the area.

This paper concerns the coefficients of the chromatic polynomial of a graph. We first report on a computational verification of the strict log-concavity conjecture for chromatic polynomials for all graphs on at most $11$ vertices, as well as for certain cubic graphs.

In the second part of the paper we give a number of conjectures and theorems regarding the behavior of the coefficients of the chromatic polynomial, in part motivated by our computations. Here our focus is on $\varepsilon(G)$, the average size of a broken-cycle-free subgraph of the graph $G$, whose behavior under edge deletion and contraction is studied.

The aim of this paper is to analyze the distribution of analytic (and signed) square roots of $\smallsha$ values on imaginary quadratic twists of elliptic curves.

Given an elliptic curve $E$ of rank zero and prime conductor $N$, there is a weight-$\frac32$ modular form $g$ associated with it such that the $d$-coefficient of $g$ is related to the value at $s=1$ of the $L$-series of the $(-d)$-quadratic twist of the elliptic curve $E$. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers $d$ the order of $\smallsha$ of the $(-d)$-quadratic twist of $E$ and analyze their distribution.

We define a combinatorial, discrete-time random walk on a closed, triangulated 3-manifold. As one varies the triangulation, keeping the number of tetrahedra fixed, the maximal mean commute time of the random walk becomes a random variable on a finite, uniform probability space of triangulations. Using computer experiments, we obtain empirical density functions for these random variables. The densities are then applied in developing Bayes-type heuristics that allow a walking entity, moving randomly in an unknown 3-manifold, to obtain probabilistic information about which manifold it might be moving in. Mean commute times are calculated via the effective electrical resistance of certain quartic graphs associated with the random walk. As a by-product, we define a topological invariant, the electrical resistance, of a 3-manifold, which we interpret as a refined complexity measure with values in the rational numbers.