Related papers: Large values of L-functions from the Selberg class

We give lower bounds of the conjectured order of magnitude for an orthogonal and a symplectic family of L-functions.

Assuming the Riemann Hypothesis, Soundararajan showed that $\displaystyle{\int_{0}^{T} \vert \zeta(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + \epsilon}}$ . His method was used by Chandee to obtain upper bounds for shifted moments of the…

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\leq 1/d$ of the L-functions of any degree $d \geq 1$ in the extended…

First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of $L$-functions from the Selberg class $\mathcal{S}$. We also study the latter abscissa inside the extended Selberg class,…

We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry…

Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some…

The theory of Selberg zeta functions is generalized to higher rank spaces. Applications towards analytic torsion numbers are given.

Let $\alpha \in (1/2,1)$ be fixed. We prove that $$ \max_{0 \leq t \leq T} |\zeta(\alpha+it)| \geq \exp\left(\frac{c_\alpha (\log T)^{1-\alpha}}{(\log \log T)^\alpha}\right) $$ for all sufficiently large $T$, where we can choose $c_\alpha =…

In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional…

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for $S(t)$. We discuss a generalization of this bound…

When it comes to partial numerical verification of the Riemann Hypothesis, one crucial part is to verify the completeness of a list of pre-computed zeros. Turing developed such a method, based on an explicit version of a theorem of…

We study the question of when two functions L_1,L_2 in the extended Selberg class are identical in terms of the zeros of L_i-h(i=1,2). Here, the meromorphic function h is called moving target. With the assumption on the growth order of h,…

Assuming the Riemann hypothesis, we prove the weak convergence of linear statistics of the zeros of L-functions towards a Gaussian field, with covariance structure corresponding to the $\HH^{1/2}$-norm of the test functions. For this…

In this paper, we demonstrate the existence of the second moment of the Selberg zeta function for a Fuchsian group of the first kind at $\sigma = 1$. The prime geodesic theorem plays a crucial role in this context. The proof extends to…

In this paper, we have proved Selberg's Central Limit Theorem for $GL(3)$ $L$-functions associated with the Hecke-Maass cusp form $f$. Moreover, we have proved the independence of the automorphic $L$-functions.

$L$-functions can be viewed axiomatically, such as in the formulation due to Selberg, or they can be seen as arising from cuspidal automorphic representations of $\textrm{GL}(n)$, as first described by Langlands. Conjecturally these two…

Motivated by the first author's earlier work in 2024, we use the resonance method to establish some Omega results for Dirichlet $L$-functions, extending the previous results.

We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood…

We introduce a "resonance" method to produce large values of $|\zeta(1/2+it)|$ and large and small central values of $L$-functions.

Low-rank approximations are popular methods to reduce the high computational cost of algorithms involving large-scale kernel matrices. The success of low-rank methods hinges on the matrix rank of the kernel matrix, and in practice, these…