Related papers: The distribution of the summatory function of the …
This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums…
Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a…
We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb Z[x]$ that is a product of at least two distinct linear…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
We formulate an abstract notion of equidistribution for families of $\lambda$-probability spaces parameterized by admissible $\mathbb{Z}$-sets. Under the assumption of equidistribution, we show that the $\sigma$-moment generating functions…
The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function,…
In this work, it is introduced a new function based on the non-trivial zeros of the Riemann-zeta function. Such function shows an interesting behavior: when the argument of the function grows, it changes from a pseudo-random behavior to a…
The aim of this work is to illustrate a conditional result involving the exponential sums over primes in short intervals under the assumption that both the Generalized Riemann Hypothesis and the Density Hypothesis for Dirichlet…
We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From…
The authors conjecture an asymptotic expression for the sixth power moment of the Riemann zeta function. They establish related results on the asymptotics of the zeta function that support the conjecture.
Lindel\"of conjectured that the Riemann zeta function $\zeta(\sigma+it)$ grows more slowly than any fixed positive power of $t$ as $t\rightarrow\infty$ when $\sigma\geq 1/2$. Hardy and Littlewood showed that this is equivalent to the…
It is known by a formula of Hasse-Sondow that the Riemann zeta function is given, for any $ s=\sigma+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty} \widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})} \sum_{k=0}^n…
Under the assumption that the distribution of a nonnegative random variable $X$ admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In particular the upper tail probability is shown to…
We study the double sum $S_\varepsilon(X)$$=$$\sum_{\substack{d,e\le X}}\frac{\mu(d)\mu(e)}{[d,e]^{1+\varepsilon}}$, which converges even in the case $\varepsilon=0$, where $\mu$ denotes the M\"obius function and $[d,e]$ is the least common…
In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Fa\'a di Bruno formula, Bell polynomials, potential polynomials,…
Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\E(z^{X_a})=f(az)/f(a)$. Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a ^{\infty}u^k\mu(du)$ for…
We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a…
Gram's Law describes a pattern that frequently occurs in the distribution of the non-trivial zeros of the Riemann zeta function along the critical line. Whenever Gram's Law holds true, it reduces the difficulty of computing the…
We report the finding of the new upper bound on the lowest positive integer $x$ for which the Mertens conjecture \begin{equation*} \left| \sum_{1 \leq n \leq x} \mu(n) \right| < \sqrt{x} \end{equation*} fails to hold: $x < \exp(1.017 \times…
We study the Tauberian relations between the moment generating function (MGF) and the complementary cumulative distribution function of a random variable whose MGF is finite only on part of the real line. We relate the right tail behavior…