Related papers: Extreme values of Dirichlet polynomials with multi…

Let $T>0$ and consider the random Dirichlet polynomial $S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$, where $(X_n)_{n}$ are i.i.d. Gaussian random variables with mean $0$ and variance $1$. We prove that the expected number of roots of…

Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $\Omega$-results for the family of quadratic Dirichlet $L$-functions $L(\sigma, \chi_d)$, where $d$ runs over all fundamental discriminants…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

We show a general lower bound for Mean-value of Dirichlet polynomials

The $n$-grid $E_n$ consists of $n$ equally spaced points in $[-1,1]$ including the endpoints $\pm 1$. The extremal polynomial $p_n^*$ is the polynomial that maximizes the uniform norm $\| p \|_{[-1,1]}$ among polynomials $p$ of degree $\leq…

We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (\pi n \tau)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $\tau$ a real quadratic…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

Fekete polynomials associated to quadratic Dirichlet characters have interesting arithmetic properties, and have been studied in many works. In this paper, we study a seemingly simpler yet rich variant: the Fekete polynomial $F_n(x) =…

Let $1 \leq p < q < \infty$. We show that \[ \sup{\frac{\left\| D\right\|_{\mathcal{H}_{q}}}{\left\| D\right\|_{\mathcal{H}_{p}}}} = \exp{\left( \frac{\log{x}}{\log{\log{x}}} \left(\log{\sqrt{\frac{q}{p}}} +…

Let $E$ be an elliptic curve over the finite field $\mathbb{F}_p$, and $P \in E(\mathbb{F}_p)$ be an $\mathbb{F}_p$-rational point. We obtain nontrivial estimates for multiplicative character sums associated with the division polynomials…

Let $\varepsilon >0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that we have almost surely, as $x \to +\infty$, $$ \sum_{n \leqslant x} f(n) \ll \sqrt{x} (\log_2 x)^{\frac{3}{4}+ \varepsilon}. $$

We obtain almost sure bounds for the weighted sum $\sum_{n \leq t} \frac{f(n)}{\sqrt{n}}$, where $f(n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated…

Fej\'er's theorem guarantees norm convergence of Ces\`aro means of Taylor partial sums in the Hardy space, whereas such convergence generally fails in weighted Dirichlet-type spaces, especially in the higher-order setting. In this paper, we…

A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…

We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are…

For $r > 0$ and integers $t \ge n > 0$, we consider the following problem: maximize the amplitude $|x_t|$ at time $t$, over all complex solutions $x = (x_0, x_1, \dots)$ of arbitrary homogeneous linear difference equations of order $n$ with…

Using the resonance method, we obtain refined estimates for joint extreme values of the Riemann zeta function at harmonic points, improving upon Levinson's 1972 results and providing new insight into the behavior of the Riemann zeta…

Let $n\in\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\mathcal{P}_n(Q)$ denote the set of polynomials over $\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding…

Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots,…

We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\mathcal{S}_d$ of $3.33d\log^2 d(1 + o(1))$ points with…