We prove that more than 41% of the zeros of the zeta function are on the critical line.Comment: 23 pages; v2: added a new paragraph on pp.1-2, minor correction

We study the 1-level density and the pair correlation of zeros of quadratic Dirichlet L-functions in function fields, as we average over the ensemble H 2g+1 of monic, square-free polynomials with coefficients in F q [x]. In the case of the 1-level density, when the Fourier transform of the test function is supported in the restricted interval ( 1 3 , 1), we compute a secondary term of size q − 4g 3 /g, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in ( 1 3 , 1 2 ), we identify another lower order term of size q − 8g 5 /g. We also compute the pair correlation, and as for the 1-level density, we detect lower order terms under certain restrictions; for example, we see a term of size q −g /g 2 when the Fourier transform is supported in ( 1 4 , 1 2 ). The 1-level density and the pair correlation allow us to obtain non-vanishing results for L( 1 2 , χ D ), as well as lower bounds for the proportion of simple zeros of this family of L-functions.

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length T 1 4 −ε. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length T 1 11 −ε were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zetafunction, zero spacing and lower bounds for moments. Along the way we obtain an asymptotic formula for a quadratic divisor problem, where the condition am 1 m 2 − bn 1 n 2 = h is summed with smooth averaging on the variables m 1 , m 2 , n 1 , n 2 , h and arbitrary weights in the average on a, b. Using Watt's work allows us to exploit all averages simultaneously. It turns out that averaging over m 1 , m 2 , n 1 , n 2 , h right away in the quadratic divisor problem simplifies considerably the combinatorics of the main terms in the twisted fourth moment.

We show that at least 1927 of the zeros of the Riemann zeta‐function are simple, assuming the Riemann hypothesis. This was previously established by Conrey, Ghosh and Gonek [Proc. London Math. Soc. 76 (1998) 497–522] under the additional assumption of the generalized Lindelöf hypothesis. We are able to remove this hypothesis by careful use of the generalized Vaughan identity.

Let χ be a primitive Dirichlet character modulo q and L(s, χ) be the Dirichlet L-function associated to χ. Using a new two-piece mollifier we show that L( 1 2 , χ) = 0 for at least 34% of the characters in the family.