Generic power series on subsets of the unit disk

In this paper, we examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $\Lambda$ in the sense of Baire category. Notably, we prove that for any open set $U$ with a non-real boundary point on the unit circle, $f(U)$ is a dense set of $\mathbb{C}$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.Comment: Wrong grant numbers fixe

Analytic properties of spherical cusp forms on GL(n)

Let $\phi$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of $\mathrm{PGL}_n$ over $\mathbb{Q}$ with Laplace eigenvalue $\lambda_{\phi}$. We establish explicit estimates for various quantities related to $\phi$ that are uniform in $\lambda_{\phi}$. This includes uniforms bounds for spherical Whittaker functions on $\mathrm{GL}_n(\mathbb{R})$, uniform bounds for the global sup-norm of $\phi$, and uniform bounds for the "essential support" of $\phi$, i.e. the region outside which it decays exponentially. The proofs combine analytic and arithmetic tools.Comment: 18 pages, LaTeX2e, submitted; v2: revised version fixing mainly (45) and its consequences; v3: revised version incorporating suggestions by the referee, e.g. Theorem 2 is now more general than before; v4: final version to appear in Journal d'Analyse Math\'ematique; v5: small corrections added in proo

Random power series near the endpoint of the convergence interval

In this paper, we are going to consider power series Sigma(infinity)(n=1)a(n)x(n), where the coefficients a(n) are chosen independently at random from a finite set with uniform distribution. We prove that if the expected value of the coefficients is 0, then lim sup(x -> 1-)Sigma(infinity)(n=1)a(n)x(n) = infinity, lim inf(x -> 1-)Sigma(infinity)(n=1)a(n)x(n) = -infinity, with probability 1. We investigate the analogous question in terms of Baire categories

Subconvexity for twisted L-functions over number fields via shifted convolution sums

Assume that π is a cuspidal automorphic GL2 representation over a number field F. Then for any Hecke character χ of conductor q, the subconvex bound L(1/2,π⊗χ)≪F,π,χ∞,εNq3/8+θ/4+ε holds for any ε>0, where θ is any constant towards the Ramanujan-Petersson conjecture (θ=7/64 is admissible). In these notes, we derive this bound from the spectral decomposition of shifted convolution sums worked out by the author in [21]

The spectral decomposition of shifted convolution sums over number fields

Let π1, π2 be cuspidal automorphic representations of GL2 over a number field F with Hecke eigenvalues λπ1(m),λπ2(m). For nonzero integers l1,l2∈F and compactly supported functions W1,W2 on F×∞, a spectral decomposition of the shifted convolution sum ∑l1t1−l2t2=q0≠t1,t2∈nλπ1(t1n−1)λπ2(t2n−1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯N(t1t2n−2)√W1(l1t1)W2(l2t2)¯¯¯¯¯¯¯¯¯¯¯¯¯ is obtained for any nonzero fractional ideal n and any nonzero element q∈n

Applications of the Kuznetsov formula on GL(3): the level aspect

We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelof on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maass forms violating the Ramanujan conjecture at finite places.Comment: The present version contains an updated version of Lemma 4 closing a small gap in the proof of Theorem

Applications of the Kuznetsov formula on GL(3) II: the level aspect

We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelöf on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maaß forms violating the Ramanujan conjecture at finite places. © 2017, Springer-Verlag GmbH Deutschland