Title: Counting zeros of the derivative of the Riemann zeta-function
Abstract: The distribution of zeros of the derivative of the Riemann zeta-function is closely related to that of zeta itself. One of the basic questions in the study of zeros is the zero-counting problem. In particular, the error terms in the zero-counting formulas are of special interest. For the Riemann zeta-function the best known bound for the error term is O(log T) due to von Mangoldt in 1905. If we assume the Riemann Hypothesis (RH), then the essentially best bound is O(log T/loglog T) due to Littlewood in 1924. For the derivative of the Riemann zeta-function Berndt proved the bound O(log T) unconditionally in 1970, and assuming RH Akatsuka proved O(log T/sqrt(loglog T)) in 2012. We show that on RH, the error term in the zero-counting formula for the derivative of Riemann zeta is O(log T/loglog T), thus it is of the same size as that for zeta itself.
Date: November 9, 2017
Abstract: Consider a system f consisting of R forms of degree d with integral coefficients. We seek to estimate the number of solutions to f=0 in integers of size B or less. A classic result of Birch (1962) answers this question when the number of variables is of size at least C(d)*R^2 for some constant C(d), and the zero set f = 0 is smooth.
We reduce the number of variables needed to C'(d)*R, and give an extension to systems of Diophantine inequalities |f| < 1 with real coefficients. Our strategy reduces the problem to an upper bound for the number of solutions to a multilinear auxiliary inequality. We relate these results to Manin's conjecture in arithmetic geometry and to Diophantine approximation on manifolds.
Date: November 23, 2017
Title: Chebyshev's bias for products of k primes and total number of prime factors in arithmetic progressions
Abstract: For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with $\omega(n)=k$ always have preference for quadratic residue classes. As an application of the method developed for settling the above problem, we consider the total number of prime factors for integers up to x among different arithmetic progressions. Greg Martin conjectured that this strong bias should be true for all x large enough. Under the same assumptions as above, we prove that this phenomenon is true for a set of x with logarithmic density 1.
Date: February 1, 2018
Title: Chebyshev's bias for products of irreducible polynomials
Abstract: Following the work of B. Cha, we adapt new results related to the Chebyshev's bias questions in the function fields setting. For any finite field F, and for any positive integer k, we study the distribution of products of k irreducible polynomials with coefficients in F in congruence classes. We obtain unconditional results for the existence of the bias. We put the emphasis on the difference from the original setting due to unexpected zeros.
Date: February 15, 2018
Abstract: We discuss the use of the large sieve for Frobenius in compatible systems along with finite monodromy groups to study the distribution of families of exponential sums. For example, this yields zero-density estimates for arguments of Kloosterman sums lying in algebraic subsets, in a general sense.
Date: March 1, 2018
Title: The sixth moment of the Riemann zeta function and ternary additive divisor sums
Abstract: Hardy and Littlewood initiated the study of the 2k-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. In this talk we consider the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula for the sixth moment. This builds on earlier work of Ivic and of Conrey-Gonek.
Date: April 12, 2018 (Note the time change 1:30-2:30 pm)
Abstract: For a number field K/Q, the class number h_K captures how far the ring of integers of K is from being a PID. The study of class numbers is a theme in number theory. In order to understand how the class number varies upon varying the number field, Siegel showed that the class number times the regulator tends to infinity in any sequence of quadratic number fields. Brauer extended this result to sequence of Galois extensions over Q. This is the Brauer-Siegel theorem. Recently, Tsfasman and Vladut conjectured a Brauer-Siegel statement for asymptotically exact sequence of number fields. In this talk, we prove the classical Brauer-Siegel and the generalized version in several unknown cases. We also provide some effective versions of Brauer-Siegel in the classical setting.
Date: May 3, 2018