Room: C630
Time:
12:00-12:50
PM
Speaker:
Nathan Ng (University of Lethbridge)
Title: Linear combinations of the zeros of the zeta
function
Abstract: The
linear independence conjecture asserts that the imaginary ordinates of
the zeros of the zeta function are linearly independent over the
rationals. This conjecture has played an important role in
several classical problems in analytic number theory including the
Mertens conjecture and the Shanks-Renyi prime number race game. I
will discuss the history of this conjecture and some preliminary
results concerning linear combinations of the zeros of zeta function.
This is joint work with Greg Martin.
Thursday,
March 4, 2010
Room: B716
Time: 4:00-4:50
PM
Speaker: Vorrapan Chandee (Stanford
University)
Title: On the correlation
of shifted values of the Riemann zeta function.
Abstract: In 2007, assuming the Riemann Hypothesis (RH),
Soundararajan proved that the 2k-th
moments of the Riemann zeta function on the critical
line is bounded by T(log T)^{k^2 + epsilon} for
every k positive real
number and every epsilon > 0.
In this talk I will generalize his methods to find
upper bounds for shifted moments.
Also I will sketch the proof how we derive their lower bounds and conjecture asymptotic formulas based on
Random matrix model, which is
analogous to Keating and Snaith's work. These upper and lower
bounds suggest that the correlation of |\zeta(1/2 +
it + i\alpha_1)| and
|\zeta(1/2 + it + i\alpha_2)| transition at
|\alpha_1 - \alpha_2| is around 1/log
T}. In particular these distribution appear independent when
|\alpha_1 - \alpha_2| is much larger than 1/log T.
Wednesday,
March 17, 2010
Room: C630
Time:
12:00-12:50
PM
Speaker:
Micah Milinovich (University of Mississippi)
Title: Central values of derivatives of Dirichlet L-functions
Abstract: It is
believed (a conjecture usually attributed to S. Chowla) that no
primitive
Dirichlet L-function vanishes at the center of the critical strip
s=1/2.
This
problem is still open. The best partial is due to H. Iwaniec and P.
Sarnak
who
have shown that at least a third of the Dirichlet L-functions, to a
large
modulus q, do not vanish at the central point. I will describe how to
modify their method and show that, for k and q large, almost all of the
values of
the k-th derivatives of primitive Dirichlet L-functions (mod q) are
nonzero
at
s=1/2. Our result compliments earlier work of P. Michel and J.
VanderKam
who
considered a similar problem. This is joint work with Hung M. Bui.
Wednesday,
March 31, 2010
Room: C630
Time:
12:00-12:50
PM
Speaker:
Habiba Kadiri (University of Lethbridge)
Title: Explicit bounds
for some prime counting functions.
Abstract: A
prime counting function assigns a weight to each prime number. For example $\pi(x)$ counts exactly the
number of prime numbers less than an arbitrarily large number $x$. It was conjectured by Gauss that
$\pi(x)$ is of the size $\frac{x}{\log x}$ as $x$ grows larger. In 1859, Riemann proposed a formula
relating prime counting functions to the zeros of a complex function
(now called the Riemann Zeta function). This innovative approach allowed
Hadamard and de la Vall\'ee Poussin to establish the Prime Number
Theorem: \[\pi(x) \sim
\frac{x}{\log x}. \] An
explicit bound for this error term has been successively investigated
by Rosser, Schoenfeld, and more recently by Dusart. I will start the talk by explaining
their method and I will then
present a new approach investigated last summer together with Laura
Faber, Allysa Lumley and Nathan Ng.
Wednesday,
April 7, 2010
Room: C630
Time:
12:00-12:50
PM
Speaker:
Habiba Kadiri (University of Lethbridge)
Title: Explicit bounds
for some prime counting functions (II)
Abstract:
This talk is a continuation of last week's talk. I
will explain how to obtain lower and upper bounds for the prime
counting function $\psi(x)$. The proof uses smooth functions and the
theory of Mellin transforms. We will also see how the previous method
of Rosser and Schoenfeld is reinterpreted through this method.