Related papers: Non-vanishing of Taylor coefficients and Poincar\'…

We prove some non-vanishing results of Hilbert Poincar\'e series. We derive these results, by showing that the Fourier coefficients of Hilbert Poincar\'e series satisfy some nice orthogonality relations for sufficiently large weight as well…

We prove a strengthening of Mui\'c's integral non-vanishing criterion for Poincar\'e series on unimodular locally compact Hausdorff groups and use it to prove a result on non-vanishing of L-functions associated to cusp forms of…

In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We examine similar congruence properties, but for the coefficients of a modified Taylor expansion about a CM point…

In this paper, we investigate several infinite products with vanishing Taylor coefficients in arihmetic progressions. These infinite products are closely related to the Rogers--Ramanujan continued fraction. Moreover, a handful of new…

We prove a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on the upper half-plane $ \mathcal H $. Moreover, we give an accompanying result on the construction of vector-valued modular forms in the…

We prove a non-vanishing result of modular L-values with quadratic twists, where the quadratic discriminants are in a short interval. Using this fact and Waldspurger's theorem, we improve the results of Balog-Ono[The chebotarev density…

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bella\"iche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier…

We prove that for any reductive group $G$ of adjoint type cuspidal automorphic twisted D-modules have non-vanishing quantum Whittaker coefficients. The argument provides a microlocal interpretation of quantum Whittaker coefficients for any…

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the…

We study when Poincar\'e series for congruence subgroups do not vanish identically. We show that almost all Poincar\'e series with suitable parameters do not vanish when either the weight $k$ or the index $m$ varies in a dyadic interval.…

In this paper, we prove modularity results of Taylor coefficients of certain non-holomorphic Jacobi forms. It is well-known that Taylor coefficients of holomorphic Jacobi forms are quasimoular forms. However recently there has been a wide…

In this semi-expository article, we discuss about the non-vanishing of the Fourier coefficients of primitive forms. Also, we shall make a note of a discrepancy in the statement of [KRW07, Lemma 2.2].

In this paper we prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) not equal to zero for n >= 1 by investigating the additive group structure attached to tau(n) with the aid of the pigeonhole principle and unique…

We prove some properties about the non-zero Taylor series coefficients $a_k$ of the Riemann xi function $\xi(s)$ at $s=\frac{1}{2}$. In particular, we present integral formulas that evaluate $a_k$ whose integrands involve a Gaussian…

We derive formulae for the calculation of Taylor coefficients of solutions to systems of Volterra integral equations, both linear and nonlinear, either without singularities or with singularities of Abel type and logarithmic type. We also…

Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} \omega_n \frac{z^n}{n!} \] with unimodular coefficients $|\omega_n|=1$. We show that either the counting function $n_F(r)$ of zeroes…

In this paper we attempt to prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) is never zero, for each n larger than zero by investigating the additive group structure attached to tau(n) with the aid of unique…

We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose…

We consider the problem of the vanishing of Poincar\'e series for congruence subgroups. Denoting by $P_{k,m,N}$ the Poincar\'e series of weight $k$ and index $m$ for the group $\Gamma_0(N)$, we show that for certain choices of parameters…

For Hamiltonian systems with degeneracy of any higher order, we study the persistence of resonant invariant tori, which as some lower-dimensional invariant tori might be elliptic, hyperbolic or of mixed types. Hence we prove a quasiperiodic…