Additive twists of Fourier coefficients of GL(3) Maass forms
We prove cancellation in a sum of Fourier coefficents of a GL(3) form $F$ twisted by additive characters, uniformly in the form $F$.
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arXiv subjects
Xiannan Li
Publications and source records attributed to Xiannan Li.
The smallest prime that does not split completely in a number field
We study the problem of bounding the least prime that does not split completely in a number field. This is a generalization of the classic problem of bounding the least quadratic non-residue. Here, we present two distinct approaches to this problem. The first is by studying the behavior of the Dedekind zeta function of the number field near 1, and the second by relating the problem to questions involving multiplicative functions. We derive the best known bounds for this problem for all number fields with degree greater than 2. We also derive the best known upper bound for the residue of the Dedekind zeta function in the case where the degree is small compared to the discriminant.
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Lower bounds for small fractional moments of Dirichlet $L$-functions
We prove a lower bound of the correct order of magnitude in the conductor aspect for small rational moments of Drichlet $L$-functions. Such bounds require new techniques, which is visible from the relationship to non-vanishing results for $L(1/2, χ)$.
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Upper bounds on L-functions at the edge of the critical strip
The problem of finding upper bounds for L-functions at the edge of the critical strip has a long and interesting history. Here, the situation for classical L-functions such as Dirichlet L-functions is relatively well understood. The reason for this is because the size of the coefficients of these L-functions is known to be small. Although L-functions are generally expected to have coefficients which are bounded by a constant at the primes, this has only been proven for a small class of familiar examples. Our main focus here is on the problem of finding upper bounds for L-functions for which we have comparatively bad bounds for the size of the coefficients.
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The argument of the Riemann $Ξ$-function off the critical line
We examine the behaviour of the zeros of the real and imaginary parts of $ξ(s)$ on the vertical line $\Re s = 1/2+λ$, for $λ\neq 0$. This can be rephrased in terms of studying the zeros of families of entire functions $A(s) = {1/2} (ξ(s+λ) + ξ(s - λ))$ and $B(s) = \frac{1}{2i} (ξ(s+λ) - ξ(s - λ))$. We will prove some unconditional analogues of results appearing in \cite{Lag}, specifically that the normalized spacings of the zeros of these functions converges to a limiting distribution consisting of equal spacings of length 1, in contrast to the expected GUE distribution for the same zeros at $λ= 0$. We will also show that, outside of a small exceptional set, the zeros of $\Re ξ(s)$ and $\Im ξ(s)$ interlace on $\Re s = 1/2+λ$. These results will depend on showing that away from the critical line, $\arg ξ(s)$ is well behaved.
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