The geometric mean is a Bernstein function

In the paper, the authors establish, by using Cauchy integral formula in the theory of complex functions, an integral representation for the geometric mean of $n$ positive numbers. From this integral representation, the geometric mean is proved to be a Bernstein function and a new proof of the well known AG inequality is provided.Comment: 10 page

Fourier Expansion of the Riemann zeta function and applications

We study the distribution of values of the Riemann zeta function $\zeta(s)$ on vertical lines $\Re s + i \mathbb{R}$, by using the theory of Hilbert space. We show among other things, that, $\zeta(s)$ has a Fourier expansion in the half-plane $\Re s \geq 1/2$ and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of $\zeta(s) - s/(s-1)$. Moreover, we discuss our results with respect to the Riemann and Lindel\"{o}f hypotheses on the growth of the Fourier coefficients.Comment: 21 page

Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process

Using the determinantal formula of Biane, Bougerol, and O'Connell, we give multitime joint probability densities to the noncolliding Brownian motion with drift, where the number of particles is finite. We study a special case such that the initial positions of particles are equidistant with a period $a$ and the values of drift coefficients are well-ordered with a scale $\sigma$. We show that, at each time $t >0$, the single-time probability density of particle system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model in the Chern-Simons theory introduced by Dolivet and Tierz. Here one-parameter extensions ($\theta$-extensions) of the Stieltjes-Wigert polynomials, which are themselves $q$-extensions of the Hermite polynomials, play an essential role. The two parameters $a$ and $\sigma$ of the process combined with time $t$ are mapped to the parameters $q$ and $\theta$ of the biorthogonal polynomials. By the transformation of normalization factor of our probability density, the partition function of the Chern-Simons matrix model is readily obtained. We study the determinantal structure of the matrix model and prove that, at each time $t >0$, the present noncolliding Brownian motion with drift is a determinantal point process, in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel. Using the obtained correlation kernel, we study time evolution of the noncolliding Brownian motion with drift.Comment: v2: REVTeX4, 34 pages, 4 figures, minor corrections made for publication in J. Math. Phy

Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function

In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions, L-series, distributions, spectral properties. These objects can be naturally defined for ?(x) as well. Despite the fact that there are various nice results about the nature of ?(x), these investigations are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals involving ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects. At the end of the paper it is shown that certain object associated with ?(x) establish a bridge between realms of imaginary and real quadratic irrationals.Comment: 34 pages, 4 figures (submitted 01/2008). Minor revisions and typos. A graph of dyadic zeta function on the critical line was added. Theorem 3 was strengthene