The AIM REU: individual projects with a common theme

We describe the main features of the Research Experience for Undergraduates Program at the American Institute of Mathematics. We also present our perspective on identifying appropriate projects for students, guiding students as they begin work, and developing a collaborative research environment while maintaining individual projects for each student. Finally, we describe some aspects of the program which others may find useful.Comment: To appear in the proceedings of the PURM conferenc

Modeling families of L-functions

We discuss the idea of a ``family of L-functions'' and describe various methods which have been used to make predictions about L-function families. The methods involve a mixture of random matrix theory and heuristics from number theory. Particular attention is paid to families of elliptic curve L-functions. We describe two random matrix models for elliptic curve families: the Independent Model and the Interaction Model

Basic analytic number theory

We give an informal introduction to the most basic techniques used to evaluate moments on the critical line of the Riemann zeta-function and to find asymptotics for sums of arithmetic functions.Comment: 11 pages, to appear in the proceedings of the school ``Recent Perspectives in Random Matrix Theory and Number Theory'' held at the Isaac Newton Institute, April 2004. Added appendix on big-O and << notatio

On the neighbor spacing of eigenvalues of unitary matrices

We describe a subtle error which can appear in numerical calculations involving the spacing statistics of eigenvalues of random unitary matrices.Comment: 4 pages. Not intended for publicatio

L-functions and higher order modular forms

It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain cases one can associate a kind of ``higher order modular form'' to such Dirichlet series. This suggests a possible approach to a proof of the conjecture.Comment: 12 pages. LaTe

Approximation by polynomials and Blaschke products having all zeros on a circle

We show that a nonvanishing analytic function on a domain in the unit disc can be approximated by (a scalar multiple of) a Blaschke product whose zeros lie on a prescribed circle enclosing the domain. We also give a new proof of the analogous classical result for polynomials. A connection is made to universality results for the Riemann zeta function

Differentiating polynomials, and zeta(2)

We study the derivatives of polynomials with equally spaced zeros and find connections to the values of the Riemann zeta-function at the positive even integers

Deformations of Maass forms

We describe numerical calculations which examine the Phillips-Sarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume Riemann surface $S$ under deformation of the surface. Our calculations indicate that if the Teichmuller space of $S$ is not trivial then each cusp form has a set of deformations under which either the cusp form remains a cusp form, or else it dissolves into a resonance whose constant term is uniformly a factor of $10^{8}$ smaller than a typical Fourier coefficient of the form. We give explicit examples of those deformations in several cases.Comment: AMSTeX, 16 pages, 13 figures. Final version, to appear in Math. Com

Converse theorems assuming a partial Euler product

Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of a particular form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $\Gamma(1)$. Weil extended this result to $\Gamma_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. Conrey and Farmer extended Hecke's result for certain small $N$, assuming that the local factors in the Euler product of $F(s)$ were of a special form. We make the same assumption on the Euler product and describe an approach to the converse theorem using certain additional assumptions. Some of the assumptions may be related to second order modular forms.Comment: 12 pages, LaTeX. Final version. To appear in The Ramanujan Journa

Differentiation Evens Out Zero Spacings

If $f$ is a polynomial with all of its roots on the real line, then the roots of the derivative $f'$ are more evenly spaced than the roots of $f$. The same holds for a real entire function of order~1 with all its zeros on a line. In particular, we show that if $f$ is entire of order~1 and has sufficient regularity in its zero spacing, then under repeated differentiation the function approaches (a change of variables from) the cosine function. We also study polynomials with all their zeros on a circle, and we find a close analogy between the two situations. This sheds light on the spacing between zeros of the Riemann zeta-function and its connection to random matrix polynomials.Comment: 22 pages, 1 figure. Final version. To appear in TAM