# Number Theory Seminar, Fall 2001

## Thursdays, 11:15-12:05, 208 Thomas

September 6              Robert Vaughan (PSU)

## Generating Functions in Additive Number Theory

September 13            Scott Parsell (PSU)

Irrational Linear Forms in Prime Variables

September 20            Mark Watkins (PSU)

The work of Alain Connes on the Riemman Hypothesis

We review Weil's "explicit formula" for the zeros of the zeta-function, and describe recent work of Connes which gives a spectral interpretation for the critical zeros, and shows the Riemann Hypothesis to be equivalent to a trace formula on the noncommutative space of Adele classes.

September 27            Robert Vaughan (PSU)

A matrix connected with the Riemann Hypothesis

October 4                    No Seminar This Week

October 11                 Helena Verrill (Hannover University)

Algorithms for higher weight modular symbols for Gamma_0(N)

Modular symbols were invented by Manin and Shokurov in order to carry out explicit computations with modular forms.  Much computational work has been done particularly in the weight 2 case by Cremona, Merel, and more recently by William Stein.  I will describe algorithms for working with modular symbols of even weight greater than two.  These methods generalise some of the methods of Cremona and Merel.  In particular I will describe "transportable" modular symbols, and an algorithm for computing the intersection pairing for modular symbols of even weight k>2.  I will also mention an application to compute the intermediate Jacobian of certain rigid Calabi-Yau threefolds.  Some of this is joint work with William Stein.

October 19                 Morley Davidson (Kent State)
NOTE:  Special Day, Time and Place: Friday, 2:30, Boucke 306.

Cyclotomic Properties of Partition Polynomials

This talk describes the results of separate joint work with George Andrews, Stephen Gagola, and Jeffrey Keen.  We define the $n^{\text{th}}$ partition polynomial $\wp_{n}(x)$ as the characteristic polynomial of Euler's linear recursion of order $\omega(n):=(3n^{2}-n)/2$ for the partition function $p(m)$ with $m < \omega(-n).$ This polynomial sequence may be generated by setting $\wp_{0}(x):=0$ and using the recursion
$$\wp_{n}(x)= x^{3n-2}\wp_{n-1}(x) + (-1)^{n-1}(x^{2n-1}-1),$$
although we have a formula based on an identity of Shanks which better clarifies the structure of $\wp_{n}(x)$.  On the basis of both symbolic algebra and numerical experiments, we arrived at a number of conjectures regarding cyclotomic properties of $\wp_{n}(x).$ While certain of the key analytic conjectures remain open, we have found proofs of several if not most of our algebraic conjectures. We will discuss three different such theorems, an example of which is the following: $x^{n}-1$ divides $\wp_{n}(x)$ for all $n;$ when $n$ is a prime exceeding 5, the quotient polynomial has maximal coefficient 2 and minimal coefficient -2, whereas if $n>1$ factors as $2^{a}3^{b},$ then these coefficients are $\pm 1. October 25 Jeff Lagarias (AT&T Labs Research) : COLLOQUIUM Some integer permutation problems from the bottom of the Erdös barrel We discuss certain permutations {a(n): n \ge 1} of the nonnegative integers which have restrictions on the greatest common divisors (a(n), a(n+1)) of consecutive terms. Such questions were first raised and studied by Paul Erdös, Robert Freud and Norbert Hegyvari in 1983. We describe work on the original problem of Erdös, Freud, and Hegyvari on making all the gcd's large, and recent work of permutations with gcd's > 1 constructed by a greedy algorithm, done jointly with Eric Rains and Neil J. A. Sloane. November 1 Yi Ouyang (University of Toronto) The Universal Norm Distribution and its Application The theory of universal ordinary distribution plays an important role in number theory, It is closely related to, the circular units in the$1$-dimensional case and the elliptic and modular units in the$2$-dimensional case. Recently Anderson proposed a double complex method to study the universal ordinary distribution. In this talk, we will generalize the universal ordinary distibution to the universal norm distribution and use Anderson's method to study it. Besides the special case of ordinary distribution, there are many other important cases of the universal norm distribution, e.g., the weight ordinary distribution, the universal Euler system. We obtain some interesting results in the index calculation. We also relate its group cohomology to the inductive procedure of the Euler system. November 8 Kevin Ford (University of Illinois at Urbana-Champaign) The prime number race We will survey many problems, results and conjectures concerning the relative magnitudes of the functions$\pi(x;q,a)$for fixed$q$. Here$\pi(x;q,a)$is the number of primes$\le x$in the progression$a$modulo$q$. It is known that for fixed$q$, all of the functions$\pi(x;q,a)$with$\text{gcd}(a,q)=1$are asymptotic to$x/(\phi(q)\log x)$, but curious inequities occur. For instance,$\pi(x;4;3) > \pi(x;4,1)$for "most" x. The behavior of such inequities is closely linked to the distribution of non-trivial zeros of Dirichlet L-functions. November 15 Michael Knapp (University of Rochester) Artin's Conjecture on Forms in Many Variables Consider a system of homogeneous polynomials in many variables with integral coefficients. A conjecture attributed to Artin states that this system will have a nontrivial simultaneous zero in p-adic integers for every prime p provided only that the number of variables is sufficiently large in terms of the degrees of the polynomials, and proposes a specific bound on how many variables suffice. We will begin this talk by discussing the extent to which the conjecture is true and mentioning some related problems. Then we will specialize to the case in which all of the polynomials are additive (ie. have no cross terms) and discuss some recent results in this situation. November 29 Eric Freeman (Institute for Advanced Study, Princeton) Systems of Diophantine equations and inequalities A Diophantine inequality is an inequality of the type$|F(\bf x)|< \epsilon$, where$F(\bf x)$is a polynomial with real coefficients, and where we seek an integral vector solution$\bf x\$. We consider certain combined systems, comprised of both Diophantine equations and inequalities. By considering these systems, we are able to show that certain systems of diagonal Diophantine inequalities of even degree have solutions.

The talk will include a presentation of the above work, but we will also spend a significant portion of the time discussing some related results and ideas.

December 6                Robert Vaughan (PSU)

Conference Report and Open Problem Session

I will start by giving a brief report on the talks given at the Richard Hall retirement meeting at the University of York in October, and then start an open problem session by describing some conjectures.  Please bring along your conjectures and be prepared to say something about them!