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Xiaosheng Wu

Publications and source records attributed to Xiaosheng Wu.

13 recordsLinked to original sources
arXiv2025

Mixed moments of twisted $L$-functions

We establish an asymptotic formula with a power-saving error term for the twisted mixed moment of Dirichlet $L$-functions and automorphic $L$-functions twisted by all primitive characters modulo $q$, valid for all admissible moduli. As a special case, this extends the asymptotic result of Blomer, Fouvry, Kowalski, Michel, and Milićević to general moduli, achieving an error term as sharp as the best bound recently proved by Khan and Zhang for prime moduli.

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arXiv2025

Upper Bounds for low moments of twisted Fourier coefficients of modular forms

For any large prime $q$, $1 \leq x \leq q$ and any real $0 \leq k \leq 1$, we prove an upper bound for the following $2k$-th moment $$\displaystyle \sum_{\substack{χ\bmod q}} \Big| \sum_{n\leq x} χ(n)λ(n)\Big|^{2k},$$ where $λ(n)$ denotes the Fourier coefficients of a fixed modular form. In particular, our result implies that $$\displaystyle \frac 1{q-1}\sum_{\substack{χ\bmod q}} \Big| \sum_{n\leq x} χ(n)λ(n)\Big|= o(\sqrt{x}),$$ when both $x$ and $q/x$ tend to infinity with $q$.

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arXiv2025

Bilinear forms with Kloosterman sums and moments of twisted L-functions

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for the second moment of (holomorphic or Maass) modular L-functions twisted with Dirichlet characters to an arbitrary large admissible modulus q. The bounds obtained are independent of the Ramanujan-Petersson conjecture and remove all factorability conditions on q in the work of Blomer, Fouvry, Kowalski, Michel, Milicevic, and Sawin.

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arXiv2025

The mollified fourth moment of Dirichlet $L$-functions

We prove an asymptotic formula with a power saving error term for the fourth moment of the family of Dirichlet $L$-functions to modulus $q$ mollified by a Dirichlet polynomial of length $q^{\frac1{22}-\ve}$, valid for all moduli $q\not\equiv2 \pmod 4$. This result was previously known only for restricted sets of moduli with smaller power savings. As a special case, when no Dirichlet polynomial is enclosed, this leads to a significant improvement on X. Wu's asymptotic evaluation of the fourth moment of Dirichlet $L$-functions at the cental point.

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arXiv2023

Binomial pmf among arithmetic progressions and sieved sets in random walks

We consider the distribution of the binomial probability mass function (pmf) among arithmetic progressions and obtain an average-type theorem. As applications, we consider the possible visits to a kind of sieved sets of integers or lattice points, by an $α$-random walker. We show that, almost surely, the asymptotic proportion of time the random walker in a sieved set of the type is equal to the density of the set, independently of $α$.

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arXiv2023

Bounds on bilinear forms with Kloosterman sums

We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet $L$-functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.

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arXiv2022

The twisted mean square and critical zeros of Dirichlet $L$-functions

In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet $L$-function with a longer mollifier, whose coefficients are also more general than before. As an application we obtain that, for every Dirichlet $L$-function, more than 41.72\% of zeros are on the critical line and more than 40.74\% of zeros are simple and on the critical line. These proportions also improve previous results which were proved only for the Riemann zeta-function.

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arXiv2022

The fourth moment of Dirichlet L-functions along the critical line

For a positive integer $q\not\equiv 2 \pmod 4$, this work considers the fourth moment of Dirichlet $L$-functions averaged over both $t\in [0,T]$ and primitive characters to modulus $q$. An asymptotic formula with a power saving from both $q$-aspect and $t$-aspect in the error term is obtained.

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arXiv2022

The fourth moment of Dirichlet $L$-functions at the central value

The asymptotic formula of the fourth moment of Dirichlet $L$-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved by M. P. Young in 2011. This work establishes this asymptotic formula for general moduli. The work relies on the study of a special divisor sum function, called $\mathcal{D}_q$-function, which plays a key role in deducing the main terms. Another key ingredient is an application of new bounds for double sums in Kloosterman sums, applied in the estimate of the error terms.

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arXiv2021

Bounds for moments of modular $L$-functions to a fixed modulus

We study the $2k$-th moment of the family of twisted modular $L$-functions to a fixed prime power modulus at the central values. We establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.

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arXiv2018

The $k$-tuple Prime Difference Champion

Let $D_{k}$ be a set with $k$ distinct elements of integers such that $d_{1}<d_{2}<\cdots<d_{k}$. We say $D_{k}^{*}$ is a $k$-tuple prime difference champion ($k$-tuple PDC) for primes $\le x$ if the set $D_{k}^{*}$ is the most probable differences among $k+1$ primes up to $x$. Unconditionally we prove that the $k$-tuple PDCs go to infinity and further have asymptotically the same number prime factors when weighted by logarithmic derivative as the porimorials. Assuming an appropriate form of the Hardy-Littlewood Prime $k$-tuple Conjecture, we obtain that the $k$-tuple PDCs are infinite square-free numbers containing any large primorial as factor when $x\rightarrow \infty$.

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arXiv2010

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.

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