# Olga BalkanovaRussian Academy of Sciences | RAS · Number theory

Olga Balkanova

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26

Publications

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Introduction

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## Publications

Publications (26)

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.

We prove an asymptotic formula for the twisted first moment of Maaß form symmetric square L-functions on the critical line and at the central point. The error term is estimated uniformly with respect to all parameters.

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.

We prove asymptotic formulas for the twisted first and second moments of Maass form symmetric square L-functions at the central point. As an application, we establish effective lower bounds on the proportion of non-zero central L-values in short intervals.

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.

We prove a spectral decomposition formula for averages of Zagier L-series in terms of moments of symmetric square L-functions associated to Maass and holomorphic cusp forms of levels 4, 16, 64.

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.

We prove an asymptotic formula for the twisted first moment of Maass form symmetric square L-functions on the critical line and at the critical point. The error term is estimated uniformly with respect to all parameters.

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.

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 Dirichle...

In this paper, we consider the family $\{L_j(s)\}_{j=1}^{\infty}$ of $L$-functions associated to an orthonormal basis $\{u_j\}_{j=1}^{\infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues $\{\lambda_j=\kappa_{j}^{2}+1/4\}_{j=1}^{\infty}$. We prove the following effective non-vanishing result: At least $40 \%$ of the c...

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

We study 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.

We prove new upper bounds for the 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...

Iwaniec and Sarnak showed that at the minimum 25% of L-values associated to holomorphic newforms of fixed even integral weight and level $N \rightarrow \infty$ do not vanish at the critical point when N is square-free and $\phi(N)\sim N$. In this paper we extend the given result to the case of prime power level $N=p^{\nu}$, $\nu\geq 2$.

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 Maa{\ss} forms of half-integral weight and the Rankin-Selberg method. The error terms a...

Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet L-series. 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-series.

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 $\Delta$ up to 2. The best previously known result, due to Iwaniec and Sarnak, was $\Delta<1$. The proof is based on a represent...

In this note we show that the methods of Motohashi and Meurman yield the same upper bound on the error term in the binary additive divisor problem. With this goal, we improve an estimate in the proof of Motohashi.

This paper studies the first moment of symmetric-square $L$-functions at the critical point in the weight aspect. Asymptotics with the best known error term $O(k^{-1/2})$ was obtained independently by Fomenko in 2005 and by Sun in 2013. We prove that there is an extra main term of size $k^{-1/2}$ in the asymptotic formula and show that the remainde...

We show that the harmonic 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 $20$ for an individual weight and at least $50$ on average. The key ingredients of our proof are the Kuznetsov convolution formula and the Liou...

We prove an asymptotic formula for the second moment of automorphic L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift and twist.

We improve the error term in the asymptotic formula for the twisted fourth moment of automorphic L functions of prime level and weight two proved by Kowalski, Michel and Vanderkam. As a consequence, we obtain a new subconvexity bound in the level aspect and improve the lower bound on proportion of simultaneous non-vanishing.

We prove an asymptotic formula for the fourth moment of automorphic
$L$-functions of level $p^{\nu}$, where $p$ is a fixed prime number and $\nu
\rightarrow \infty$. 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 Du...