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Jeffrey Hoffstein

Publications and source records attributed to Jeffrey Hoffstein.

4 recordsLinked to original sources
arXiv2026

Non-orthogonality of the cubic and quartic large sieves via Rankin-Selberg

We show unconditionally that the cubic and quartic large sieves are not perfectly orthogonal. The main obstruction to perfect orthogonality comes from the bias exhibited by Gauss sums. Our proof requires two main inputs: a Lindel\"{o}f-on-average upper bound for the second moment of Kubota's Dirichlet series, and a tight average lower bound for the Fourier coefficients of a certain Rankin-Selberg convolution of metaplectic theta functions. The latter input is particularly important in the quartic case, where much less is known about Fourier coefficients of metaplectic theta functions. To establish both of these inputs, we adapt a Rankin-Selberg regularization method due to Zagier (1981). In addition to the cubic and quartic cases considered in this paper, we expect that the family of Hecke characters of each fixed order $n \geq 3$ over a number field $K \supset \mathbb{Q}(\zeta_n)$ is not perfectly orthogonal. We provide a precise conjecture for the operator norm of these ensembles for each $n$.

math.NT↗
arXiv2023

The shifted convolution L-function for Maass forms

Let $\Phi_1,\Phi_2$ be Maass forms for $\text{SL}(2,\mathbb Z)$ with Fourier coefficients $C_1(n),C_2(n)$. For a positive integer $h$ the meromorphic continuation and growth in $s\in\mathbb C$ (away from poles) of the shifted convolution L-function $$L_h(s,{\Phi_1,\Phi_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ is obtained. For ${\rm Re}(s) > 0$ it is shown that the only poles are possible simple poles at $\frac{1}{2} \pm ir_k$, where $\tfrac14+r_k^2$ are eigenvalues of the Laplacian. As an application we obtain, for $T\to\infty$, the asymptotic formula \begin{align*} & \underset{n \neq 0,-h}{\sum_{\sqrt{|n (n + h)|}<T} } \hskip-5pt{C_1(n) C_2(n + h)} \left(\text{log}\Big(\tfrac{T}{\sqrt{|n (n + h)|}}\,\Big)\right)^{\frac{3}{2} + \varepsilon} \hskip-7pt =\; f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \cdot T^{\frac{1}{2}} \; + \; \mathcal O\left( h^{1-\varepsilon} T^\varepsilon + h^{1 + \varepsilon} T^{-1 - \varepsilon} \right), \end{align*} where the function $f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T)$ is given as an explicit spectral sum that satisfies the bound $f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \ll h^{\theta + \varepsilon}$. We also obtain a sharp bound for the above shifted convolution sum with sharp cutoff, i.e., without the smoothing weight $\log(*)^{\frac32+\varepsilon}$ with uniformity in the $h$ aspect. Specifically, we show that for $h < x^{\frac{1}{2} - \varepsilon}$, \[ {\sum_{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}\theta + \varepsilon}x^{\frac{2}{3} (1 + \theta) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2\theta + \varepsilon}. \]

math.NT↗
arXiv2001

Multiple Dirichlet series and moments of zeta and L-functions

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.

math.NT↗