Related papers: Extremal bounds for Dirichlet polynomials with ran…

The article studies the almost surely asymptotics of extreme values $\bar{\xi}_n = \max_{1\leq i \leq n} \xi_i$, where $ \xi , \xi_1 , \xi_2 , \ldots$ are discrete identically distributed random variables. One of the main results on this…

An explicit Dirichlet series is obtained, which represents an analytic function of $s$ in the half-plane $\Re s>1/2$ except for having simple poles at points $s_j$ that correspond to exceptional eigenvalues $\lambda_j$ of the non-Euclidean…

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density…

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an…

Let $F(\sigma)=\sum_{n=1}^\infty \frac{X_n}{n^\sigma}$ be a random Dirichlet series where $(X_n)_{n\in\mathbb{N}}$ are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of…

We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log…

Let $(\xi_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |\xi_i|\big/…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve…

Non-separable $D-$dimensional partial differential Schr\"{o}dinger equations are considered at $D=2$ and $D=3$, with the even-parity local potentials $V(x,y,\ldots)$ which are polynomials of degree four (cusp catastrophe resembling case)…

Conditionally on the Riemann hypothesis we prove asymptotic formulae for mean values of various long Dirichlet polynomials involving the von Mangoldt function. Our results avoid the use of correlation sum estimates although in addition to…

Let $(d_n)$ be a sequence of positive numbers and let $(X_n)$ be a sequence of positive independent random variables. We provide an upper bound for the deviation between the distribution of the mantissaes of $(X_n^{d_n})$ and the Benford's…

In this paper, we considier the limiting distribution of the maximum interpoint Euclidean distance $M_n=\max _{1 \leq i<j \leq n}\left\|\boldsymbol{X}_i-\boldsymbol{X}_j\right\|$, where $\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots,…

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…

A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows…

We consider the Norros-Reittu random graph $NR_n(\textbf{w})$, where edges are present independently but edge probabilities are moderated by vertex weights, and use probabilistic arguments based on martingales to analyse the component sizes…

We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is…

Let $n$ be a non-zero integer. A set $S$ of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \in S$ with $a \neq b$. It is of special interest to estimate the quantity $M_n$, the…

Let X be a generalised symmetrised Dirichlet random vector in R^k, and let u_n be thresholds such that P{X> u_n} tends to 0 as n goes infinity. In this paper we derive an exact asymptotic expansion of P{X> u_n} assuming that the associated…

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet…

Let $X_1, \ldots, X_n$ be independent random points drawn from an absolutely continuous probability measure with density $f$ in $\mathbb{R}^d$. Under mild conditions on $f$, we derive a Poisson limit theorem for the number of large…