- Tuesday,
**January 18**, 2005.

**Harold Diamond, UIUC.**

**Title: Cosh-like arithmetic functions having an average value.** - Thursday,
**January 20**, 2005.

**Florin Boca, UIUC.**

**Title: The correlations of Farey fractions.**

**Abstract:**It will be proved that all correlation measures of the sequence of Farey fractions exist. The pair correlation function is explicitly computed and shows an intermediate behaviour between the Poisson and the GUE correlations. This is joint work with A. Zaharescu. - Tuesday,
**January 25**, 2005.

**Kevin Ford, UIUC.**

**Title: Integers covered by systems of congruences with distinct moduli.**

**Abstract:**A covering system is a finite set of arithmetic progressions with distinct moduli >1 whose union is all the integers. Erdos conjectured that there are covering systems whose smallest moduli is arbitrarily large. We consider the following related problem: If N is a large integer and K>1, what is the densest union of arithmetic progressions with distinct moduli in [N,KN]? We show that if K is a bit smaller than a power of N, then the desity of the union of progressions cannot be much more than 1-1/K. In particular, a covering system with distinct moduli in [N,KN] cannot exist. This is joint work with Michael Filaseta, Sergei Konyagin, Carl Pomerance and Gang Yu. - Thursday,
**January 27**, 2005.

**Maria Sabitova, University of Pennsylvania.**

**Title: Root numbers of abelian varieties.**

**Abstract:**We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves over the field of rational numbers. Our result applies to abelian varieties over number fields. Namely, under certain conditions which naturally extend the conditions used by D. Rohrlich, we show that the root number W(A,\tau) associated to an abelian variety A over a number field F and a complex finite-dimensional irreducible representation \tau of the absolute Galois group of F with real-valued character is equal to 1. In the case where the ground field is Q, we show that our result is consistent with the refined version of the conjecture of Birch and Swinnerton-Dyer. - Tuesday,
**February 1**, 2005.

**Alexandru Zaharescu, UIUC.**

**Title: Lehmer numbers and Lehmer points.**

**Abstract:**We discuss the definition of Lehmer numbers and Lehmer points, and present some asymptotic results on the number of such points. - Tuesday,
**February 8**, 2005.

**Lenny Fukshansky, Texas A&M.**

**Title: Effective structure theorems for quadratic spaces and their isometries**

**Abstract:**A classical theorem of Witt states that a bilinear space can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. I will discuss the existence of such a decomposition of bounded height for a symmetric bilinear space over a number field, where all bounds on height are explicit. I will also talk about an effective version of Cartan-Dieudonne?? theorem on representation of an isometry of a regular symmetric bilinear space as a product of reflections. Finally, if time permits, I will show a special version of Siegel's Lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. - Thursday,
**February 10**, 2005.

**Emre Alkan, UIUC.**

**Title: Diophantine Approximation with mild divisibility constraints.**

**Abstract:**I will present a survey on the history of Diophantine approximation problems using special subsets of integers and also talk about recent work on this jointly with A. Zaharescu and G. Harman - Tuesday,
**February 15**, 2005.

**Stephanie Treneer, UIUC.**

**Title: Congruences for the coefficients of weakly holomorphic modular forms, I.**

**Abstract:**Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomena is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form. In particular, we give congruences for a wide class of partitions functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level. Tuesday's talk will consist of an introduction to the problem, a statement of the main theorems, and a discussion of the two applications. Thursday we will prove the main theorems. - Thursday,
**February 17**, 2005.

**Stephanie Treneer, UIUC.**

**Title: Congruences for the coefficients of weakly holomorphic modular forms, II.**

**Abstract:**Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomena is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form. In particular, we give congruences for a wide class of partitions functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level. Tuesday's talk will consist of an introduction to the problem, a statement of the main theorems, and a discussion of the two applications. Thursday we will prove the main theorems. - Tuesday,
**February 22**, 2005.

**Timothy Kilbourn, UIUC.**

**Title: An Extension of the Supercongruence for Apery Numbers.**

**Abstract:**In 1987, Beukers proved a mod p congruence between the coefficients of a certain modular form and the Apery numbers. He conjectured the existence of a ``supercongruence'' mod p^2; this was proved by Ahlgren and Ono in 2000. In this talk we prove an extension of this congruence mod p^3. The proof involves the modularity of a Calabi-Yau threefold, the Gross-Koblitz formula, the p-adic gamma function, and some interesting combinatorics. - Thursday,
**February 24**, 2005.

**Michael Drmota, Technical University, Wien.**

**Title: Digital expansions, exponential sums and central limit theorems.**

**Abstract:**The purpose of this talk is to present recent results on the distribution of the values of the ($q$-ary) sum of digits function $s_q(n)$, where we mainly focus on two problems, the joint distribution $(s_{q_1}(n),s_{q_2}(n))$ of two different expansions and on the distribution of the sum-of-digits function of squares: $s_q(n^2)$.

Interestingly there are strong relations to "classical analytic number theory". For example, one can use Baker's theorem on linear forms (and exponential sums) to prove that the pairs $(s_{q_1}(n),s_{q_2}(n))$, $n - Tuesday,
**March 1**, 2005.

**Heini Halberstam, UIUC.**

**Title: On some very old theorems of Erdos** - Thursday,
**March 3**, 2005.

**Paul Bateman, UIUC.**

**Title: The Dirichlet series for the reciprocal of the Riemann zeta function**

**Abstract:**For real values of s greater than 1, we compare the partial sums of the series $\sum \mu(n)n^{-s}$ with its sum $1/\zeta(s)$, where $\zeta$ denotes the Riemann zeta-function. Specifically, we ask: For what values of N does the difference $$\sum_{n=1}^N \mu(n)n^{-s} - 1/\zeta(s) $$ have a fixed sign for all s in $(1,\infty)$. The talk is related to my recent problem in the American Mathematical Monthly, which deals with some simple cases. - Tuesday,
**March 8**, 2005.

**Ben Howard, University of Chicago.**

**Title: The Iwasawa theoretic Gross-Zagier theorem.**

**Abstract:**The Gross-Zagier theorem relates the central value of the derivative of the L-function of an elliptic curve to the Neron-Tate height of a special point, a Heegner point, in the Mordell-Weil group. Combining this result with work of Kolyvagin gives some of the strongest known results in the direction of the Birch and Swinnerton-Dyer conjecture. In this talk we will explain an Iwasawa theoretic analogue (conjectured by Mazur and Rubin) of the Gross-Zagier theorem. The result relates towers of Heegner points in a Z_p-extension to the derivative of a two-variable p-adic L-function. - Tuesday,
**March 15**, 2005.

**A. Raghuram, University of Iowa.**

**Title: Special values of L-functions.**

**Abstract:**This talk will be an introduction to Deligne's conjectures on the special values of the symmetric power L-functions associated to a holomorphic cuspform. The latter half of the talk will be an account of some work in progress, which is joint work with Freydoon Shahidi, toward the special values of the symmetric fourth power L-functions. - Thursday,
**March 17**, 2005.

**Bruce Berndt, UIUC.**

**Title: Ramanujan's 40 Identities for the Rogers-Ramanujan Functions.**

**Abstract:**Ramanujan's list of 40 identities was found in the Oxford University Library by Bryan Birch and published in 1975. The list is in the handwriting of G. N. Watson who privately held the manuscript for many years and evidently lost Ramanujan's original list. We provide a survey of the methods used over the years to prove the identities, with emphasis on recent work. This is joint work with Geumlan Choi, Youn-Seo Choi, Heekyoung Hahn, Boon Pin Yeap, Ae Ja Yee, Hamza Yesilyurt, and Jinhee Yi. - Tuesday,
**March 29**, 2005.

**Bruce Reznick, UIUC.**

**Title: A computational version of a theorem of Polya.**

**Abstract:**In 1928, Polya proved that if p is a homogeneous polynomial in n variables which is strictly positive on the unit simplex, then for sufficiently large N, all the coefficients of (x1+...+xn)^N * p are non-negative. Vicki Powers and I proved in 2001 an explicit estimate on N, which is sharp in at least one special case. This theorem is becoming popular in certain circles of mathematical optimization. All proofs are elementary. - Thursday,
**March 31**, 2005.

**Florin Boca, UIUC.**

**Title: The Dirichlet series of a certain arithmetic function associated with the continued fraction algorithm.** - Tuesday,
**April 5**, 2005.

**Ken Stolarsky, UIUC.**

**Title: A curious confluence of three problems in the analytic theory of polynomials.**

**Abstract:**How a polynomial behaves at critical points (e.g., max-min theory) and how the symmetry of a polynomial relates to the distribution of its zeros are two major problem areas in the study of polynomials. A more esoteric problem is to what extent a polynomial and some of its derivatives can all be composite (say without having multiple zeros). Certain multivariable polynomials over the rationals shall be displayed that seem to be remarkable (at least to me) with respect to each of these three problem areas. They are somehow related to Shabat polynomials. - Thursday,
**April 7**, 2005.

**Ben Brubaker, Stanford University.**

**Title: The form of Fourier coefficients of Eisenstein Series.**

**Abstract:**We will begin with a review of ordinary spectral Eisenstein series on GL(2) and its Fourier coefficients and then describe a new class of Eisenstein series associated to covers of semi-simple Lie groups (called metaplectic groups since Weil originally worked over the symplectic group Sp(2n)). The Fourier (or more properly Whittaker) coefficients of these Eisenstein series have extremely rich arithmetic information, e.g. they can contain Dirichlet L-functions or interesting combinations of Gauss sums. Hence analytic information about the Eisenstein series can be translated into information about the special values of L-functions, etc. We'll present some recent examples of this and give a combinatorial approach to dealing with these objects that avoids much of the complication with computing the Fourier-Whittaker coefficients directly. - Tuesday,
**April 12**, 2005.

**Kevin Ford, UIUC.**

**Title: On the difference between a number and its inverse modulo n.**

**Abstract:**We study the distribution of the function M(n) = max(|a-b|: 0 < a, b < n and ab=1 (mod n)) and its connection to divisor problems. In particular, we examine what can be said (i) for infinitely many n, (ii) for infinitely many prime n, and (iii) for almost all n. - Thursday,
**April 14**, 2005.

**Various faculty and graduate students in number theory. UIUC.**

**Title: 10 minute talks.**

**Abstract:**Volunteers from the regular seminar group will offer short talks in the area of general number theory, roughly ten minutes in length. - Tuesday,
**April 19**, 2005.

**O-Yeat Chan, UIUC.**

**Title: A Table of Values in Ramanujan's Lost Notebook.**

**Abstract:**On pages 179 and 180 of his Lost Notebook, Ramanujan lists a table of values related to the crank generating function. In this talk, we shall use the circle method to prove the completeness of this table, and, as a bonus, prove 2 conjectures of Andrews and Lewis regarding cranks modulo 3 and 4. - Thursday,
**April 21**, 2005.

**Andrew Booker, University of Michigan.**

**Title: Converse theorems and Artin's conjecture.**

**Abstract:**Given a finite Galois extension of the rationals and a representation of its Galois group, Artin defined an L-function which he conjectured to have analytic continuation to the complex plane and satisfy a functional equation. All progress to date on this conjecture has come through the related Langlands' program which anticipates that Artin's L-functions agree with those associated to automorphic forms. I will show that if Artin's conjecture is true for a given 2-dimensional representation then so is Langlands' conjecture. The technique used in the proof may be extended to give a Weil-type converse theorem. If time permits, I will discuss generalizations to number fields. - Tuesday,
**April 26**, 2005.

**Matt Boylan, UIUC.**

**Title: Parity of the partition function.**

**Abstract:**Let p(n) denote the ordinary parition function. In 1966 Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N = r (mod t) with p(N) even, and infinitely many integers M = r (mod t) with p(M) odd. Using classical facts about modular forms mod 2, we prove the conjecture for every arithmetic progression r (mod t) where t is a power of 2. This is joint work with Ken Ono from 2001. - Thursday,
**April 28**, 2005.

**Ling Long, Iowa State University.**

**Title: Eisenstein series and representing natural numbers as sum of integer squares.**

**Abstract:**In this talk we are going to address two questions. We will give a short proof of Milne's formulae for sums of $4n^2$ and $4n^2+4n$ integer squares using the theory of modular forms, in particular Eisenstein series. This work was done jointly with Yifan Yang. In the second part, we will address a question regarding the zeros of some Eisenstein series used in the first part of the talk. - Tuesday,
**May 3**, 2005.

**Kathrin Bringmann, University of Wisconsin.**

**Title: Traces of Singular Moduli on Hilbert Modular Surfaces.**

**Abstract:**Suppose that $p\equiv 1\pmod 4$ is a prime, and that $\Op$ is the ring of integers of $K:=\Q(\sqrt{p})$. A classical result of Hirzebruch and Zagier asserts that certain generating functions for the intersection numbers of Hirzebruch-Zagier divisors on the Hilbert modular surface $(\h\times \h)/\SL_2(\Op)$ are weight $2$ holomorphic modular forms. Using recent work of Bruinier and Funke, we show that the generating functions of traces of singular moduli over these intersection points are often weakly holomorphic weight $2$ modular forms. For the singular moduli of $J_1(z)=j(z)-744$, we explicitly determine these generating functions using classical Weber functions, and we factorize their ``norms" as products of Hilbert class polynomials. - Thursday,
**May 5**, 2005.

**Paul van Wamelen, Lousiana State University.**

**Title: Computing with the Jacobian of a Hyperelliptic Curve.**

**Abstract:**We will define the algebraic and analytic Jacobians of a hyperelliptic curve and explain how to go from the one to the other. We will demonstrate functionalities in the computer algebra system MAGMA for working with these objects.