Shifted moments of modular $L$-functions to a fixed level
Peng Gao, Liangyi Zhao
本文在广义黎曼猜想下，研究了模$L$-函数在固定素数水平下的移位矩的上界问题。
我们在广义黎曼猜想下，建立了模$L$-函数在固定素数水平下的移位矩的上界。
We establish upper bounds for shifted moments of modular $L$-functions to a fixed prime level under the generalized Riemann hypothesis.
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The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.
Special Values of Shifted Dirichlet Series from an Adjoint Map on Almost Holomorphic Modular Forms
该研究找到几乎全纯模形式空间上某映射的伴随映射，将其类比拟模形式空间上的广义Ramanujan-Serre导数，借此明确求出相关移位狄利克雷级数的特殊值并证明其有理性结果。
Hoffstein and Hulse recently introduced the notion of shifted convolution Dirichlet series for pairs of modular forms $f_1$ and $f_2$. The second two authors investigated certain special values of symmetrized sums of such functions, numbers which are generally expected to be mysterious transcendental numbers. They proved that the generating functions of these values in $h$-aspect are linear combinations of mixed mock modular forms and quasimodular forms. Here we examine the special cases when $f_1=f_2$ where, in addition, there is a prime $p$ for which $p^2$ divides the level. We prove that th…
Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes
研究Salié和的移位双线性和的上界，并分析移位素数的模平方根分布，揭示$b\not \equiv 0 \bmod q$时的新困难。
The shifted convolution problem for Fourier coefficients of Siegel modular forms of degree $2$
本文提出了一种基于自守诠释的新方法，首次对更高秩群上的移位卷积和给出了非平凡估计，通过估计塞尔曼泊松级数的傅里叶系数实现。
The shifted fourth moment of modular form L-functions in the weight aspect
本文研究了模形式L函数在重量方面的移位第四矩，并将其与黎曼ζ函数和自守L函数的矩联系起来，探讨了主项结构以推广到更高矩。
In this paper, a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols is proved.
Using explicit constructions of the Weierstrass mock modular form, we offer a closed formula for generating the values of shifted convolution $L$-values for certain elliptic curves that can be computed to arbitrary precision. These identities provide a surprising relation between weight $2$ newforms and shifted convolution $L$-values.
We give a characterisation of the field into which quotients of values of L-functions associated to a cusp form belong. The construction involves shifted convolution series of divisor sums and to establish it we combine parts of F. Brown's technique to study multiple modular values with the properties of a double Eisentein series previously studied by the author and C. O'Sullivan.
Assuming the Riemann Hypothesis, Soundararajan showed that $\displaystyle{\int_{0}^{T} \vert ζ(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + ε}}$ . His method was used by Chandee to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$- functions which allow us to derive upper bounds for moments of theta functions.
Lower bounds for shifted moments of Dirichlet $L$-functions of fixed modulus
本文在广义黎曼猜想下，建立了固定模Dirichlet $L$-函数双平移矩的精确下界。
Shifted moments of quadratic Dirichlet $L$-functions
本文在广义黎曼假设下，研究了二次狄利克雷 $L$-函数的移位矩并给出了上界，同时应用该结果推导了二次狄利克雷特征和的矩的界限。
Holomorphic projection for sesquiharmonic Maass forms
本文研究半调和Maass形式的全纯投影，给出涉及实二次类数与theta函数乘积的数值表达，并分析移位卷积L级数及其和的上界。
Bounds for Moments of Twisted Fourier coefficients of Modular Forms
本文在广义黎曼假设下，为模L函数的移位矩及二次扭矩建立上界，并据此推导出模形式傅里叶系数与狄利克雷字符扭和之矩的界。
Hoffstein and Hulse defined the shifted convolution series of two cusp forms by "shifting" the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of certain values of these series and prove a polynomial bound as h approaches infinity. Our method relies on a result of Mertens and Ono, who showed that these values are Fourier coefficients of mixed mock modular forms.
In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution $L$-functions by defining shifted convolution $L$-functions. We investigate symmetrized versions of their functions. Under certain mild conditions, we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and "mixed mock modular" forms.
Upper bounds for shifted moments of quadratic Dirichlet $L$-functions over function fields
本文在函数域上建立了二次Dirichlet $L$-函数平移矩的锐利上界，并将其应用于证明二次Dirichlet特征和矩的若干界。
Upper bounds for shifted moments of Dirichlet $L$-functions to a fixed modulus over function fields
本文建立了函数域上固定模 Dirichlet $L$-函数族平移矩的精确上界，并以此推导出 Dirichlet 特征和矩的上界。
Shifted moments of the Riemann zeta function
本文证明黎曼假设蕴含Chandee关于黎曼zeta函数平移矩的猜想，证明基于Harper关于临界线上$2k$次矩精确上界的思想。
Shifted moments of cubic and quartic Dirichlet $L$-functions
本文在广义黎曼猜想下，建立了三次和四次狄利克雷L函数位移矩的上界，并应用此结果给出了三次和四次狄利克雷特征和的矩界。
We prove the asymptotic formula for the fourth moment of automorphic $L$-functions of level $p^ν$, where $p$ is a fixed prime number and $ν\rightarrow \infty$. This paper is a continuation of work by Rouymi, who computed asymptotics of the first three moments at prime power level, and a generalization of results obtained for prime level by Duke, Friedlander & Iwaniec and Kowalski, Michel & Vanderkam.
We prove an asymptotic formula for the second moment (up to height $T$) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to $T^{2-ε}$, for any $ε>0.$
Upper bound for the moment of shifted values of cubic $L$-functions over function fields
该研究针对函数域上三次L-函数的平移值相关性，推导了其矩的上界，结果适用于q≡2 mod 3的非库默情形，q≡1 mod 3的库默情形可类似处理。
Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions
本文在GRH和GRC假设下，研究自守形式与Eisentein级数的混合四阶矩，揭示非均匀分布现象并提出相关猜想。
The shifted prime-divisor function over shifted primes
本文研究了移位素数除数函数 $ω_a^*$ 在移位素数上的矩，证明了第一矩的渐近公式及第二矩的上下界，揭示了其在移位素数与自然数上平均行为的相似性，并提出了关于第二矩的新猜想。
Sharp lower bounds for shifted moments of Dedekind zeta functions
在假设GRH成立的前提下，通过双陪集稳定子相关因子，证明了戴德金ζ函数平移矩的精确下界，结合对应上界确定了伽罗瓦与非伽罗瓦场平移矩的量级。
For $f$ a primitive holomorphic cusp form of even weight $k \geq 4$, level $N$, and $χ$ a Dirichlet character mod $Q$ with $(Q,N)=1$, we establish a new hybrid subconvexity bound for $L(1/2 + it, f_χ)$, which improves upon all known hybrid bounds. This is done via amplification and taking advantage of a shifted convolution sum of two variables defined and analyzed in a recent paper of Hoffstein and Hulse.
We investigate the first and second moments of shifted convolutions of the generalised divisor function $d_3(n)$.
In this paper we work over $Γ_0(N)$, for any $N$ and write the spectral moment of a product of two distinct Rankin-Selberg convolutions at a general point on the critical line $\frac{1}{2}+it$ as a main term plus a sharp error term in the $t$ aspect and the spectral aspect. As a result we obtain hybrid Weyl type subconvexity results in the $t$ and spectral aspects. Also, for fixed modular forms $f$, $g$ of even weight $k\geq 4$ we show there exists a Maass cusp form $u_j$ such that $L(1/2, f\times u_j)$, $L(1/2, g\times u_j)$ are simultaneous non-zero.
The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form $p_{S_1}(n-H) = p_{S_2}(n)$ for all $n$ in some arithmetic progression. Several identities of this type have been discovered, including two infinite families found by Alladi. In this paper, we use the theory of modular functions to determine the necessary and sufficient conditions for such an …
Shifted-prime divisors
本文研究由 Prachar 提出的移位素数因数计数函数 $\omega^*(n)$ 的高阶矩及相关问题。
These are notes for four lectures given at the 2010 CIMPA Research School "Automorphic Forms and L-functions" in Weihai, China. The lectures focused on a Burgess-like subconvex bound for twisted Hilbert modular L-functions published jointly with Valentin Blomer in the same year. They discussed the proof in some detail, especially how spectral theory can be used to estimate the relevant shifted convolution sums efficiently. They also discussed briefly an application for the number of representations by a totally positive ternary quadratic form over a totally real number field.
We prove an asymptotic for the second moment of quadratic twists of a modular $L$-function. This was previously known conditionally on GRH by the work of Soundararajan and Young.
Moments of the shifted prime divisor function
本文证明了移位素数除数函数的矩的渐近公式，证实了 Fan 和 Pomerance 的猜想，其证明结合了最小公倍数的组合恒等式与数论解析工具。
Bloch-Okounkov studied certain functions on partitions $f$ called shifted symmetric polynomials. They showed that certain $q$-series arising from these functions (the so-called \emph{$q$-brackets} $\left<f\right>_q$) are quasimodular forms. We revisit a family of such functions, denoted $Q_k$, and study the $p$-adic properties of their $q$-brackets. To do this, we define regularized versions $Q_k^{(p)}$ for primes $p.$ We also use Jacobi forms to show that the $\left<Q_k^{(p)}\right>_q$ are quasimodular and find explicit expressions for them in terms of the $\left<Q_k\right>_q$.
Shifted second moment of Gaussian Hecke $L$-functions $L(s,λ^k)$
该研究建立了高斯角Hecke L函数L(s,λ^k)的光滑加权移位二阶矩的一致渐近公式，通过转化非对角贡献为Weyl和等方法完成证明，所得公式与L函数比率猜想的四交换预测一致，误差为O_{Φ,ε}(K^{1/2+ε})。
On certain kernel functions and shifted convolution sums of Hecke eigenvalues
本文针对全模群上偶整权重的归一化本原全纯尖点形式的第j次对称幂L-函数系数，研究以k-full核函数为权的移位卷积和的行为，属于数论领域的解析数论研究。
We study the second moment of the central values of quadratic twists of a modular $L$-function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.
Bounds for integral moments of quadratic twists of Fourier coefficients of modular forms
本文在广义黎曼猜想下给出模形式傅里叶系数二次扭特征和的积分矩上界，并证明其最优性。
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 $0 \leq k \leq 1$.
Lower bounds for shifted moments of the Riemann zeta function
该研究在黎曼猜想成立的假设下，证明了Chandee提出的黎曼ζ函数平移矩的对应下界，补充了作者此前得到的上界结果，完善了这类矩的界值研究。
Moment of Derivatives of Quadratic Twists of Modular $L$-Functions
本文研究了两个不同模$L$-函数二次扭导数矩的渐进行为，首次在不依赖GRH条件下给出证明。
Lower bounds for moments of quadratic twists of modular $L$-functions
该论文证明了模$L$-函数二次扭曲在中心点的$2k$-阶矩（$k \geq 1/2$）的尖锐下界。
We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q, with an error term that has a close to optimal power savings in the exponent. However, one of the contributions to the main term is a special value of a shifted double Dirichlet series. We show that this special value is small on average, and obtain a corresponding estimate for a mean value of the second moment over Q. This mean value is non-zero even when applied to a product of two distinct L-ser…
We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Appli…
In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment} proved that $\int_{0}^T |ζ(1/2 + it)|^{2k} dt \ll_{k, ε} T(\log T)^{k^2 + ε}$ for every $k$ positive real number and every $ε> 0.$ In this paper we generalized his methods to find upper bounds for shifted moments. We also obtained their lower bounds and conjectured asymptotic formulas based on Random matrix model, which is analogous to Keating and Snaith's work. These upper and lower bounds suggest that the correlation of $|ζ(\h + it + iα_1)|$ and $|ζ(\h + it + iα_2)|$ transition at $|α_1 - α_2| \approx \frac{1}{\log T}…
Twisted second moment of modular $L$-functions to a fixed modulus
本文渐近研究了模 $L$ 函数族关于固定模的扭曲二阶矩，并以此为基础，为这些函数在临界线上的 $2k$ 阶矩确立了特定范围内的精确上下界。
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise input of the conjectural formula for the classical shifted convolution problem for divisor sums so as to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^2$ with divisor functions as coefficients.
We prove a sharp bound on the fifth moment of modular L-functions of fixed small weight, and large prime level.
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^3$ with divisor functions as coefficients.
The first moment of quadratic twists of modular $L$-functions
本文基于 Young 的工作，研究了模 $L$ 函数二次扭曲及其一阶导数的平滑第一矩，并给出了误差项为 $O(X^{\frac{1}{2} + \varepsilon})$ 的渐近公式。
The second moment of derivatives of quadratic twists of modular $L$-functions
本文证明了二次模形式 $L$-函数二次扭曲导数的二阶矩渐进行式，包含三个主导项，改进了 Kumar 等人的先前结果，方法基于 Li 建立的大筛不等式及大位移扭曲 $L$-函数渐进行式计算。
In this work we provide a meromorphic continuation in three complex variables of two types of triple shifted convolution sums of Fourier coefficients of holomorphic cusp forms. The foundations of this construction are based in the continuation of the spectral expansion of a special truncated Poincaré series recently developed by Jeffrey Hoffstein. As a result we are able to produce previously unstudied and nontrivial asymptotics of truncated shifted sums which we expect to correspond to off-diagonal terms in the third moment of automorphic L-functions.
Shifted second moment of the Riemann zeta function and a Fourier type kernel
本文在更大区域上利用 Riemann-Siegel 公式计算黎曼 zeta 函数移位二阶矩，并证明其作为虚移位函数具有 Fourier-Cauchy 型核行为，类似正交函数。
In this series of papers we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper completes the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.
Low moments of random multiplicative functions twisted by Fourier coefficients of modular forms
研究随机乘法函数与模形式傅里叶系数的矩性质，确定在特定范围内的矩阶大小。
Modularity of moments of reciprocal sums for partitions into distinct parts
本文研究了互异分拆中部分倒数 $k$ 次幂之和的生成函数的模性，证明其与 Maass Eisenstein 级数和半调和 Maass 形式相关。
We consider negative moments of quadratic Dirichlet $L$--functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $\mathbb{F}_q[x]$, we obtain an asymptotic formula for the $k^{\text{th}}$ shifted negative moment of $L(1/2+β,χ_D)$, in certain ranges of $β$ (for example, when roughly $β\gg \log g/g $ and $k<1$). We also obtain non-trivial upper bounds for the $k^{\text{th}}$ shifted negative moment when $\log(1/β) \ll \log g$. Previously, almost sharp upper bounds were obtained in \cite{ratios} in the range $β\gg g^{-\frac{1}{2k}+ε}$.
Lower Bounds on High Moments of Twisted Fourier coefficients of Modular Forms
本文研究了模形式扭曲Fourier系数的高次矩问题，针对大素数q和实数k≥2，证明了2k次矩的下界，并在广义Riemann假设下证明了该界在常数因子范围内是精确的。
We define two-parameter generalizations of Andrews' $(k+1)$-marked odd Durfee symbols and $2k$th symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When $k=0$ we obtain families of modular forms and mock modular forms. When $k \geq 1$, we find quasimodular forms and quasimock modular forms.
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