862 research outputs found

### 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

### 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

### 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

### 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

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