The fourth moment of Dirichlet L-functions

In this paper, we study the fourth moment of Dirichlet $L$-functions averaged over primitive characters to modulus $q$ and over $t\in [0,T]$, for general $q$ and $T$. When $T\gg q^\varepsilon$ is not too small with respect to $q$, we obtain an asymptotic formula with a power savings in the error term. In addition, when $q$ is a prime, the weak condition $T\gg q^\varepsilon$ is unnecessary, as well as a more sharp error has been obtained. These improvements also benefit from an extension of Young's classical work [The fourth moment of Dirichlet L-functions, Ann. of Math. (2) 173 (2011), no. 1, 1-50] for prime $q$.Comment: 63 pages, any comments are welcome. We fix some minor typos in English in this versio

On gaps between zeros of the Riemann zeta function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.Comment: submitted for publication in January 201

The twisted mean square and critical zeros of Dirichlet LL-functions

The asymptotic formula for mean square of the Riemann zeta-function times a Dirichlet polynomial of length $T^\theta$ is proved when $\theta<17/33$ and $\theta<4/7$ for a special form of the coefficient, while for a general Dirichlet $L$-function, it is only proved when $\theta<1/2$, without any special better result, by Bauer\cite{Bau} in 2000. This is due to the additional Dirichlet character contained in the coefficient, which causes error terms harder to control. In this work, we prove that a general Dirichlet $L$-function has the same results as the Riemann zeta-function. A more general form of the coefficient than one in Conrey\cite{Con-More} is also obtained for the $\theta<4/7$ case. As an application we obtain that, for every Dirichlet $L$-function, more than .4172 zeros are on the critical line and more than .4074 zeros are on the critical line and simple.Comment: 33 page