Related papers: The Error Term in the Sato-Tate Conjecture

Suppose that the Riemann hypothesis is false and $\rho_{*} = 1/2 + \eta_{*} + i \gamma_{*}$, $\eta_{*} > 0$, is a nontrivial zero of the Riemann $\zeta$-function off the critical line. Under the negation of the Riemann hypothesis for the…

The $2 q$-th pseudomoment $\Psi_{2q,\alpha}(x)$ of the $\alpha$-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $\zeta^\alpha$ on the critical line. Using probabilistic methods of…

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq…

For a prime $p$, we consider Kloosterman sums $$ K_{p}(a) = \sum_{x\in \F_p^*} \exp(2 \pi i (x + ax^{-1})/p), \qquad a \in \F_p^*, $$ over a finite field of $p$ elements. It is well known that due to results of Deligne, Katz and Sarnak, the…

Let $f$ be a holomorphic modular form of prime level $p$ and trivial nebentypus. We show that there exists a computable $\delta>0$, such that $$ L\left(\tfrac{1}{2},\mathrm{Sym}^2 f\right)\ll p^{\tfrac{1}{2}-\delta}, $$ with the implied…

We prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ of…

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of…

Let $\big(\mathcal{S}_n^{\alpha,\kappa,\mathfrak{r}}(z)\big)_{n=1}^\infty$ be a sequence of the largest possible integer intervals, such that…

For an $L^2$-normalized holomorphic newform $f$ of weight $k$ on a hyperbolic surface of volume $V$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb{Q}$, we prove the sup-norm estimate \[ \|…

Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1<p<\infty. It is known that if A and its adjoint A^* both satisfy square function estimates \bignorm{\bigl(\int_{0}^{\infty}| A^{1/2} T_t(x)|^2\, dt\,\bigr)^{1/2}_{Lp} \lesssim…

For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the adele ring $\mathbb{A}$, we study the distribution of angles $\theta_\pi(\mathfrak{p})$ coming from Satake parameters corresponding to unramified…

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…

We address the analysis of the following problem: given a real H\"older potential $f$ defined on the Bernoulli space and $\mu_f$ its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by…

Let $p_n$ denote the $n$-th prime. In 2000, Panaitopol established the inequality $p_1 \cdots p_n > p_{n+1}^{n - \pi(n)}$ for all $n \geq 2$, where $\pi(x)$ is the prime counting function. In 2021, Yang and Liao refined this by introducing…

We examine the size of $E_{2}(T)$, the error term in the asymptotic formula for $\int_{0}^{T} |\zeta(1/2 + it)|^{4}\, dt$ where $\zeta(s)$ is the Riemann zeta-function. We make improvements in the powers of $\log T$ in the known bounds for…

Fix $\delta\in(0,1]$, $\sigma_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $\pi_{\mathcal P}(x)$ satisfies an…

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\GL_2(\A_F)$, $F$ a totally real…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

The $2$kth pseudomoments of the Riemann zeta function $\zeta(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $\zeta(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like…

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.