Abstract:
We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the Riemann zeta function whose spacing is 2.9125 times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.

Abstract:
In this article we exhibit small and large values of $\zeta'(\rho)$ by applying Soundararajan's resonance method. Our results assume the Riemann hypothesis.

Abstract:
In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $\zeta'(\rho)$ where $\zeta(s)$ is the Riemann zeta function and $\rho$ is a non-trivial zero of the Riemann zeta function.

Abstract:
Discrete moments of the Riemann zeta function were studied by Gonek and Hejhal in the 1980's. They independently formulated a conjecture concerning the size of these moments. In 1999, Hughes, Keating, and O'Connell, by employing a random matrix model, made this conjecture more precise. Subject to the Riemann hypothesis, we establish upper and lower bounds of the correct order of magnitude in the case of the fourth moment.

Abstract:
Let the summatory function of the M\"{o}bius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. The main results shown are that the weak Mertens conjecture and the existence of a limiting distribution of $e^{-y/2}M(e^{y})$ are consequences of the aforementioned conjectures. By probabilistic techniques, we present an argument that suggests $M(x)$ grows as large positive and large negative as a constant times $\pm \sqrt{x} (\log \log \log x)^{{5/4}}$ infinitely often, thus providing evidence for an unpublished conjecture of Gonek's.

Abstract:
This article studies the zeros of Dedekind zeta functions. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of the Deuring-Heilbronn phenomenon. In addition, we obtain an explicit bound for the number of zeros in a box.

Abstract:
Let $L(s,\chi)$ be a fixed Dirichlet $L$-function. Given a vertical arithmetic progression of $T$ points on the line $\Re(s)=1/2$, we show that $\gg T \log T$ of them are not zeros of $L(s,\chi)$. This result provides some theoretical evidence towards the conjecture that all ordinates of zeros of Dirichlet $L$-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of $L(s,\chi)$.

Abstract:
Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan that provides analogous bounds for moments of $L$-functions at the central point, averaged over families.

Abstract:
Assuming the generalized Riemann hypothesis, we prove quantitative estimates for the number of simple zeros on the critical line for the L-functions attached to classical holomorphic newforms.

Abstract:
We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function averaged over the zeros of the zeta-function that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.