H. M. Bui

More than 41% of the zeros of the zeta function are on the critical line

We prove that at least 41.05% of the zeros of the Riemann zeta function are on the critical line.

A note on the gaps between consecutive 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 most 0.5155 times the average spacing and that infinitely often they differ by at least 2.6950 times the average spacing.

On simple zeros of the Riemann zeta-function

We show that at least 19/27 of the zeros of the Riemann zeta-function are simple, assuming the Riemann Hypothesis (RH). This was previously established by Conrey, Ghosh and Gonek [Proc. London Math. Soc. 76 (1998), 497–522] under the additional assumption of the Generalised Lindelof Hypothesis (GLH). We are able to remove this hypothesis by careful use of the generalised Vaughan identity.

NON-VANISHING OF DIRICHLET L-FUNCTIONS AT THE CENTRAL POINT

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(½, χ) ≠ 0 for at least 34% of the characters in the family.

CENTRAL VALUES OF DERIVATIVES OF DIRICHLET L-FUNCTIONS

Let

A note on the second moment of automorphic L-functions

\mathscr{C}_{q}^{+}

Gaps between zeros of Dedekind zeta-functions of quadratic number fields. II

be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method, we show that L(k)(½, χ) ≠ 0 for almost all the characters

On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class

\chi\in\mathscr{C}_{q}^{+}

A note on the fourth moment of Dirichlet

when k and q are large. Here L(s, χ) is the Dirichlet L-function associated to the character χ.

L

We obtain the formula for the twisted harmonic second moment of the L -functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean-value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet L -functions.