Number Theory and Combinatorics Seminar

Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010

Wednesday,
February 24, 2010

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.

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.