Related papers: Probabilistic interpretation of the M\"obius funct…

We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in…

Let m>=1 be an arbitrary fixed integer and let N_m(x) count the number of odd integers u<=x such that the order of 2 modulo u is not divisible by m. In case m is prime estimates for N_m(x) were given by H. Mueller that were subsequently…

The wrapped normal distribution arises when a the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. At first look, evaluation of its probability density function appears tedious as an…

In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent…

A Friedman number is a positive integer which is the result of an expression combining all of its own digits by use of the four basic operations, exponentiation and digit concatenation. A "nice" Friedman number is a Friedman number for…

The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of the coherent state representation of…

Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected…

Let $\pi$ be the occupancy density of an obliquely reflected Brownian motion in the half plane and let ($\rho$, $\alpha$) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of…

We prove that the distribution of the product of two correlated normal random variables with arbitrary means and arbitrary variances is infinitely divisible. We also obtain exact formulas for the probability density function of the sum of…

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main…

We obtain the asymptotic behavior of hole probability for random holomorphic sections on a compact Riemann surface with respect to the hole size.

In this paper we consider the distribution of the zeros of a real random Bargmann-Fock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations…

Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free $\Delta=D/16$, where D is the discriminant, by T.D. Browning and…

A new class of identical particles which may exhibit both Bose and Fermi statistics with respective probabilities $p_b$ and $p_f$ is introduced. Such an uncertainity may be either an intrinsic property of a particle or can be viewed as an…

Many physical parameters in astrophysics are derived using the ratios of two observed quantities. If the relative uncertainties on measurements are small enough, uncertainties can be propagated analytically using simplifying assumptions,…

We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for…

It is known that the M\"obius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in \N}$, for a…

A number is said to be $y$-smooth if all of its prime factors are less than or equal to $y.$ For all $17/30<\theta\leq 1,$ we show that the density of $y$-smooth numbers in the short interval $[x,x+x^{\theta}]$ is asymptotically equal to…

We introduce M\"obius functions of higher rank, a new class of arithmetic functions so that the classical M\"obius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free…