Large gaps between consecutive zeros of the Riemann zeta-function. II

Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing

Twists of automorphic L-functions at the central point

We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character \chi modulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.\chi,s) do not vanish at s=1/2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.\chi,s), and from that derive an upper bound for the average analytic rank of the studied L-functions

A note on the second moment of automorphic L-functions

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.Comment: 9 page

Gaps between zeros of the derivative of the Riemann \xi-function

Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of \xi'(s). We prove that a positive proportion of gaps are less than 0.796 times the average spacing and, in the other direction, a positive proportion of gaps are greater than 1.18 times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than 0.7203 (1.5, respectively).Comment: 15 page

Robust Dialog State Tracking for Large Ontologies

The Dialog State Tracking Challenge 4 (DSTC 4) differentiates itself from the previous three editions as follows: the number of slot-value pairs present in the ontology is much larger, no spoken language understanding output is given, and utterances are labeled at the subdialog level. This paper describes a novel dialog state tracking method designed to work robustly under these conditions, using elaborate string matching, coreference resolution tailored for dialogs and a few other improvements. The method can correctly identify many values that are not explicitly present in the utterance. On the final evaluation, our method came in first among 7 competing teams and 24 entries. The F1-score achieved by our method was 9 and 7 percentage points higher than that of the runner-up for the utterance-level evaluation and for the subdialog-level evaluation, respectively.Comment: Paper accepted at IWSDS 201

Gaps between zeros of the Riemann zeta-function

We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist)