Related papers: Simple zeros of modular L-functions

This theorem is based on holomorphy of studied functions and the fact that near a singularity point the real part of some rational function can take an arbitrary preassigned value.

It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li…

The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of…

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…

We study, on average over f, zeros of the L-functions of primitive weight two forms of level q (fixed). We prove, on the one hand, density theorems for the zeros (similar to the results of Bombieri, Jutila, Motohashi, Selberg in the case of…

A proof for the original Riemann hypothesis is proposed based on the infinite Hadamard product representation for the Riemann zeta function and later generalized to Dirichlet L-functions. The extension of the hypothesis to other functions…

In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) =…

We apply the calculus of variations to construct a new sequence of linear combinations of derivatives of the Riemann $\zeta$-function adapted to Levinson's method, which yield a positive proportion of zeros of the $\zeta$-function on the…

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…

We investigate a dynamical basis for the Riemann hypothesis (RH) that the non-trivial zeros of the Riemann zeta function lie on the critical line x = 1/2. In the process we graphically explore, in as rich a way as possible, the diversity of…

We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence…

Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…

We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter $\tau$, we introduce a conceptual proof by applying Painlev\'{e} VI\…

In this paper, we investigate the zeros near the critical line of linear combinations of $L$-functions belonging to a large class, which conjecturally contains all $L$-functions arising from automorphic representations on $\text{GL}(n)$.…

The first nontrivial zeroes of the Riemann $\zeta$ function are $\approx 1/2+\pm14.13472i$. We investigate the question of whether or not any other L-function has a higher lowest zero. To do so we try to quantify the notion that the…

In this paper, we exhibit upper and lower bounds with explicit constants for some objects related to entire $L$-functions in the critical strip, under the generalized Riemann hypothesis. The examples include the entire Dirichlet…

In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip.

In this paper, we study weighted low-lying zeros of spinor and standard $L$-functions attached to degree 2 Siegel modular forms. We show the symmetry type of weighted low-lying zeros of spinor $L$-functions is symplectic, for test functions…

Applying Littlewood's lemma in connection to Riemann's Hypothesis and exploiting the symmetry of Riemann's $xi$ function we show that almost all nontrivial Riemann's Zeta zeros are on the critical line.

Let $F$ be a linear combination of $N\geq 1$ Dirichlet $L$-functions attached to even (or odd) primitive characters with the same modulus. Selberg proved that a positive proportion of non-trivial zeros of $F$ lie on the critical line. Our…