arXiv subjects
Publications and source records attributed to Fatma Çiçek.
We compute the twisted first and second moments of the shifted central values of the Rankin-Selberg $L$-functions given by $L\left(\frac12+ω, f\otimes g\right)$ as $f$ varies over primitive forms of prime power level $p^ν$ with $ν\geq 3$. Here $ω$ is a bounded shift and $g$ is a fixed primitive form of level relatively prime to $p$.
In the function field setting with a fixed characteristic, it was proven by the second and third authors that the values $\log \big|L\big(\frac12, χ_D\big)\big|$ as $D$ varies over monic and square-free polynomials are asymptotically Gaussian distributed on the assumption of a low lying zeros hypothesis as the degree of $D$ tends to $\infty$. For real distinct shifts $t_j$ all of microscopic size or all of nonmicroscopic size relative to the genus, we consider linear combinations of $\log\big|L\big(\frac12+it_j, χ_D\big)\big|$ with real coefficients, and separately, of $\arg L\big(\frac12+it_j, χ_D\big).$ We provide estimates for their distribution functions under the low lying zeros hypothesis. We similarly study distribution functions of linear combinations of $\log\big|L\big(\frac12+it_j, E\otimes χ_D\big)\big|$, and separately $\arg L\big(\frac12+it_j, E\otimesχ_D\big)$, for quadratic twists of elliptic curves $E$ with root number one as the conductor gets large. As an application of these results, we prove a central limit theorem for the fluctuation of the number of nontrivial zeros of such $L$-functions from its mean, and thus recover previous results by Faifman and Rudnick. Correlations of such fluctuations are in harmony with the results of Bourgade, Coram and Diaconis, and Wieand for zeros of the Riemann zeta function and for eigenangles of unitary random matrices.
Let $L(s, χ_1), \ldots, L(s, χ_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log|L(ρ,χ_1)|+\dots+a_N\log|L(ρ,χ_N)|$ has an approximately normal distribution as $T\to \infty $ with mean $0$ and variance $ \tfrac12 \big({a_1}^2+\dots+{a_N}^2\big)\log\log T.$ Here $a_1, a_2, \ldots, a_N \in \mathbb{R}$, and $ρ$ runs over the nontrivial zeros of the zeta function with $0< \Imρ\leq T$. From this we deduce that the vectors $\big(\log|L(ρ,χ_1)|/\sqrt{ \frac12 \log\log T}, \ldots, \log|L(ρ,χ_N)|/\sqrt{\frac12 \log\log T}\,\big)$ have approximately an $N$-variate normal distribution whose components are approximately mutually independent as $T\to \infty$. We apply these results to study the proportion of the $ρ$ that are zeros or $a$-values of linear combinations of the form $c_1 L(ρ, χ_1)+ \cdots + c_N L(ρ, χ_N)$ with complex $c_i$ as coefficients.
On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $\frac12+iγ$ of the Riemann zeta function, we show that the sequence \[ Γ_{[a, b]} =\Bigg\{ γ: γ>0 \quad \mbox{and} \quad \frac{ \log\big(| ζ^{(m_γ)} (\frac12+ iγ) | / (\logγ)^{m_γ}\big)}{\sqrt{\frac12\log\logγ}} \in [a, b] \Bigg\}, \] where the $γ$ are arranged in increasing order, is uniformly distributed modulo one. Here $a$ and $b$ are real numbers with $a<b$, and $m_γ$ denotes the multiplicity of the zero $\frac12+iγ$. The same result holds when the $γ$'s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $γ(\log T)/2π$ with $γ\in Γ_{[a, b]}$ and $0<γ\leq T$.
The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(π,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $π$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $π$ of $G_2(\mathbb{A})$ that corresponds to a modular form $φ$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $Λ(π,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $φ$ is nonzero, we prove the functional equation $Λ(π,\operatorname{Std},s) = Λ(π,\operatorname{Std},1-s)$. Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal.
The purpose of this paper is to generalize our earlier work on the logarithm of the Riemann zeta-function to linear combinations of logarithms of primitive Dirichlet $L$-functions with constant real coefficients. Under the assumption of suitable hypotheses, we prove that as $T\to \infty $, a sequence of the form $(a_1\log|L(ρ,χ_n)|+\dots+a_1\log|L(ρ,χ_n)|)$ has an approximate Gaussian distribution with mean $0$ and variance $ \tfrac{1}{2}\big({a_1}^2+\dots+{a_n}^2 \big)\log\log T$. Here $a_1, \dots, a_n \in \mathbb{R}$, each of the $χ_i$ is a primitive Dirichlet character modulo $M_i$ with $M_i\leq T$, and $0< \operatorname{Im}ρ\leq T$ where $ρ$ runs over nontrivial zeros of the zeta-function. From the proof of this result, we also derive the independence of the distributions of sequences $(\log|L(ρ,χ_1)|), \dots, (\log|L(ρ,χ_n)|)$ provided that they are suitably normalized.