Correlations of values of random diagonal forms

We study the value distribution of diagonal forms in k variables and degree d with random real coefficients and positive integer variables, normalized so that mean spacing is one. We show that the l-correlation of almost all such forms is Poissonian when k is large enough depending on l and d.Comment: 28 page

Simultaneous extreme values of zeta and L-functions

We show that distinct primitive L-functions can achieve extreme values simultaneously on the critical line. Our proof uses a modification of the resonance method and can be applied to establish simultaneous extreme central values of L-functions in families.Comment: 37 page

Simultaneous large values and dependence of Dirichlet LL-functions in the critical strip

We consider the joint value distribution of Dirichlet $L$-functions in the critical strip $\frac{1}{2} < \sigma < 1$. We show that the values of distinct Dirichlet $L$-functions are dependent in the sense that they do not behave like independently distributed random variables and they prevent each other from obtaining large values. Nevertheless, we show that distinct Dirichlet $L$-functions can achieve large values simultaneously infinitely often.Comment: 23 page

Extreme values of L-functions

The value distribution of the Riemann zeta function $\zeta(s)$ is a classical question. Despite the fact that values of $\zeta(s)$ are approximately Gaussian distributed, $\zeta(s)$ can be very large for infinitely many $s$ as $|\Im s|\rightarrow\infty$. Exponential sums and random matrix theory have been extensively employed to study the behaviour of extreme values of $\zeta(s)$. This thesis is focused on extreme values of $L$-functions using the resonance method together with recent developments on greatest common divisor sums. This thesis consists of five chapters. The first chapter gives some history and recent progress on the extreme values of $L$-functions in the critical strip. In Chapter \ref{chap2}, we consider large values of the Dedekind zeta function $\zeta_K(s)$ in the critical strip, where $K$ is an arbitrary number field . We present two different approaches to the problem: one is to use Phragmen-Lindel\"of principle, and the other is to use the convolution method. This is based on joint work with S. Baluyot and A. Zaharescu. In Chapter \ref{chap3}, we focus on large values of degree $1$ $L$-functions in the Selberg class in relation to other $L$-functions. It is believed that values of distinct primitive $L$-functions behave like independent random variables. For example, if we let $\rho$ denote a zero of a Dirichlet $L$-function, we can ask what the behaviour of $\zeta(\rho)$ is. The method is based on the study of simple zeros of $\zeta(s)$, and the results apply to general $L$-functions in the Selberg class satisfying appropriate conditions. In Chapter \ref{chap4}, we discuss small values of the derivative of the Dedekind zeta function. It is believed that $\zeta_K'(\rho)$ cannot be zero from the Grand Simplicity Conjecture. We show that $\zeta_K'(\rho)$ can be very small for infinitely many $\rho$. This is a generalization of a result of N. Ng to number fields beyond \QQ. The last chapter lists all the references