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Dmitry Frolenkov

Publications and source records attributed to Dmitry Frolenkov.

At least 19 recordsLinked to original sources
arXiv2026

Moments and Non-Vanishing of Maass Form Symmetric Square L-Functions in Short Intervals

Recently, Li obtained a mean Lindel\"of estimate for the cubic moment of the central values of Maass form symmetric square $L$-functions over short intervals $(T-H, T+H)$ of length $H \ge T^{18/19+\epsilon}$. We improve this result by showing that the estimate holds for $H \gg T^{6/7+\epsilon}$. The key ingredient in our proof is a new asymptotic formula for the second twisted moment of Maass symmetric square $L$-functions. Based on this formula, we also improve the lower bound for the proportion of non-vanishing central $L$-values in short intervals. Previously, even under the assumption of the Lindel\"of hypothesis for Dirichlet $L$-functions, the proportion of non-vanishing values in intervals of length $H = T^{\beta}$ was only known to be at least $\frac{3\beta-1}{4}$. We establish an unconditional lower bound of $\frac{7\beta-2}{8}$.

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arXiv2026

Non-vanishing of symmetric square $L$-functions in the weight aspect

We prove a new asymptotic formula for the second moment of symmetric square L-functions in the weight aspect on average. This result implies that the associated L-function is non-vanishing at the central point for at least 76.69% of holomorphic Hecke cusp forms of bounded weight, improving the previous bound of 70.37%.

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arXiv2024

Hybrid subconvexity for Maass form symmetric-square $L$-functions

Recently R. Khan and M. Young proved a mean Lindel\"{o}f estimate for the second moment of Maass form symmetric-square $L$-functions $L(\text{sym}^2 u_{j},1/2+it)$ on the short interval of length $G\gg |t_j|^{1+\epsilon}/t^{2/3}$, where $t_j$ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for $L(\text{sym}^2 u_{j},1/2+it)$ as long as $|t_j|^{6/7+\delta} \ll t<(2-\delta)|t_j|$. We obtain a mean Lindel\"{o}f estimate for the same moment in shorter intervals, namely for $G\gg |t_j|^{1+\epsilon}/t$. As a corollary, we prove a subconvexity estimate for $L(\text{sym}^2 u_{j},1/2+it)$ on the interval $|t_j|^{2/3+\delta}\ll t\ll |t_j|^{6/7-\delta}$.

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arXiv2024

The shifted fourth moment of modular form L-functions in the weight aspect

We prove a reciprocity type formula for the fourth moment of L-functions associated to holomorphic primitive cusp forms of level one and large weight which relates it to the eighth moment of the Riemann zeta function and the dual weighted fourth moments of automorphic L-functions (both holomorphic and Maass). The main objective of the paper is to study the structure of the main term for possible generalization of the method to higher moments.

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arXiv2021

On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case

We prove a Weyl-type subconvex bound for cube-free level Hecke characters over totally real number fields. Our proof relies on an explicit inversion to Motohashi's formula. Schwartz functions of various kinds and the invariance of the relevant Motohashi's distributions discovered in a previous paper play central roles.

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arXiv2020

The second moment of symmetric square L-functions over Gaussian integers

We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog-Biro-Cherubini-Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.

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arXiv2019

Prime geodesics and averages of the Zagier $L$-series

The Zagier $L$-series encode data of real quadratic fields. We study the average size of these $L$-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

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arXiv2019

Prime geodesic theorem for the Picard manifold

Let $Γ=PSL(2,Z[i])$ be the Picard group and $H^3$ be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient $Γ\setminus H^3$, called the Picard manifold, obtaining an error term of size $O(X^{3/2+θ/2+ε})$, where $θ$ denotes a subconvexity exponent for quadratic Dirichlet $L$-functions defined over Gaussian integers.

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arXiv2018

Moments of L-functions and Liouville-Green method

We show that the percentage of primitive forms of level one and weight $4k\rightarrow \infty$ for which the associated $L$-function at the central point is no less than $(\log{k})^{-2}$ is at least 20%. The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville-Green method.

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arXiv2018

Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions

Let $ \mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k \in N $. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(\mathfrak{f}, 1/2) $ and the symmetric square $L$-function $ L(sym^2\mathfrak{f}, 1/2)$, relating it to the dual mixed moment of the double Dirichlet series and the Riemann zeta function weighted by the ${}_3F_{2}$ hypergeometric function. Analysing the corresponding special functions by the means of the Liouville-Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded by $\log^3k$.

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arXiv2018

Sums of Kloosterman sums in the prime geodesic theorem

We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.

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arXiv2018

Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

For $Γ$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=Γ\backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_Γ(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For the Picard manifold, $Γ=\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_Γ(X)=O(X^{5/3+ε})$, to $E_Γ(X)=O(X^{13/8+ε})$. In the process we obtain a mean subconvexity estimate for the Rankin-Selberg $L$-function attached to Maass-Hecke cusp forms. We also investigate the second moment of $E_Γ(X)$ for a general cofinite group $Γ$, and show that it is bounded by $O(X^{16/5+ε})$.

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arXiv2018

Bounds for a spectral exponential sum

We prove new upper bounds for a spectral exponential sum by refining the process by which one evaluates mean values of $L$-functions multiplied by an oscillating function. In particular, we introduce a method which is capable of taking into consideration the oscillatory behaviour of the function. This gives an improvement of the result of Luo and Sarnak when $T\geq X^{1/6+2θ/3}$. Furthermore, this proves the conjecture of Petridis and Risager in some ranges. Finally, this allows obtaining a new proof of the Soundararajan-Young error estimate in the prime geodesic theorem.

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arXiv2018

Convolution formula for the sums of generalized Dirichlet L-functions

Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square $L$-functions and an asymptotic expansion for the average of central values of generalized Dirichlet $L$-functions.

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arXiv2017

A Mean Value Result for a Product of GL(2) and GL(3) L-Functions

In this paper various analytic techniques are com- bined in order to study the average of a product of a Hecke L- function and a symmetric square L-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin-Selberg method. The error terms are bounded using the Liouville-Green approximation.

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arXiv2017

The first moment of cusp form L-functions in weight aspect on average

We study the asymptotic behaviour of the twisted first moment of central $L$-values associated to cusp forms in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of mollifier $Δ$ up to 2. The best previously known result, due to Iwaniec and Sarnak, was $Δ<1$. The proof is based on a representation formula for the error in terms of Legendre polynomials.

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