Related papers: Simple zeros of $\mathrm{GL}(2)$ $L$-functions

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $\chi$ be a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^{\kappa}$ with $p$ prime and $\kappa>12$. A subconvex bound for the…

Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire $L$-function of real archimedian type has a zero in the interval $\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\approx 14.13$ corresponds to the first…

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also…

For $f$ a cuspidal modular form for the group $\Gamma_0(N)$ of integral or half-integral weight, $N$ a multiple of $4$ in case the weight is half-integral, we study the zeros of the $L$-function attached to $f$ twisted by an additive…

For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $\gamma_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…

We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field,…

In this paper, we study lower bounds of a general family of $L$-functions on the $1$-line. More precisely, we show that for any $F(s)$ in this family, there exists arbitrary large $t$ such that $F(1+it)\geq e^{\gamma_F} (\log_2 t + \log_3…

We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet $L$-functions of modulus $q$ at height $T$. To do this, we derive an asymptotic for the twisted second moment of Dirichlet…

Suppose that $k$ and $N$ are positive integers. Let $f$ be a newform on $\Gamma_0(N)$ of weight $k$ with $L$-function $L_f(s)$. Previous works have studied the zeros of the period polynomial $r_f(z)$, which is a generating function for the…

We provide a short proof of a conjecture of Davila and Kenter concerning a lower bound on the zero forcing number $Z(G)$ of a graph $G$. More specifically, we show that $Z(G)\geq (g-2)(\delta-2)+2$ for every graph $G$ of girth $g$ at least…

We show that if $A$ is a finite set of non-negative integers then the number of zeros of the function \[ f_A(\theta) = \sum_{a \in A} \cos(a\theta), \] in $[0,2\pi]$, is at least $(\log \log \log |A|)^{1/2-\varepsilon}$. This gives the…

Let $F$ be a number field. Let $\pi_1,\pi_2$ be cuspidal automorphic representations of $GL_2(\mathbb{A}_F)$, and let $\pi$ be a cuspidal automorphic representation of either $GL_2(\mathbb{A}_F)$ or $GL_3(\mathbb{A}_F)$. When…

In this paper, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg L-functions. One of the main new input is a substantial improvement of the admissible length of the mollifier which is…

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$,…

Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2, \mathbb{Z})$ . Let $\chi$ be a primitive character of modulus $P$. We shall prove the following results: 1. Suppose $P = p^r$, where $p$ is a prime and…

Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll…

Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg…

Let $O_L$ be the ring of integers of a number field $L$. Write $q = e^{2 \pi i z}$, and suppose that $$f(z) = \sum_{n \gg - \infty}^{\infty} a_f(n) q^n \in M_{k}^{!}(\operatorname{SL}_2(\mathbb{Z})) \cap O_L[[q]]$$ is a weakly holomorphic…

Fix $g$ a self-dual Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $f$ be a holomorphic newform of prime level $q$ and fixed weight. Conditional on a lower bound for a short sum of squares of Fourier coefficients of $f$, we prove a…

Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet $L$-functions are simple. This improves on earlier work of \"{O}zl\"{u}k which gives a proportion of at…