arXiv subjects

Djurdje Cvijović

Publications and source records attributed to Djurdje Cvijović.

7 recordsLinked to original sources

arXiv2010

Proof of the Borwein-Broadhurst conjecture for a dilogarithmic integral arising in quantum field theory

Borwein and Broadhurst, using experimental-mathematics techniques, in 1998 identified numerous hyperbolic 3-manifolds whose volumes are rationally related to values of various Dirichlet L series $\textup{L}_{d}(s)$. In particular, in the simplest case of an ideal tetrahedron in hyperbolic space, they conjectured that a dilogarithmic integral representing the volume equals to $\textup{L}_{-7}(2)$. Here we have provided a formal proof of this conjecture which has been recently numerically verified (to at least 19,995 digits, using 45 minutes on 1024 processors) in cutting-edge computing experiments. The proof essentially relies on the results of Zagier on the formula for the value of Dedekind zeta function $ζ_{\mathbb{K}}(2)$ for an arbitrary field $\mathbb{K}$.

math-ph↗

arXiv2010

Polypseudologarithms revisited

Lee, in a series of papers, described a unified formulation of the statistical thermodynamics of ideal quantum gases in terms of the polylogarithm functions, $\textup{Li}_{s} (z)$. It is aimed here to investigate the functions $\textup{Li}_{s} (z),$ for $s = 0, -1, -2, ...,$ which are, following Lee, referred to as the polypseudologarithms (or polypseudologs) of order $n$. Various known results regarding polypseudologs, mainly obtained in widely differing contexts and currently scattered throughout the literature, have been brought together along with many new results and insights and they all have been proved in a simple and unified manner. In addition, a new general explicit closed-form formula for these functions involving the Carlitz--Scoville higher tangent numbers has been established.

math-ph↗

arXiv2009

Integral representations of the Legendre chi function

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $χ_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta function $ζ(2 n+1)$ and the Dirichlet beta function $β(2 n)$ ,$ n\in\mathbb{N}$, are direct consequence of these representations.

math.CA↗

arXiv2009

Closed-form formulae for the derivatives of trigonometric functions at rational multiples of $π$

In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of all derivatives of four trigonometric functions at rational multiples of $π$ can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials. In addition, some particular cases are considered.

math.CA↗

arXiv2009

New integral representations of the polylogarithm function

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function $Li_(z)$. The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm for any complex z for which $|z|$ < 1. Two are valid for all complex s, whenever $\Re(s)>1$ . The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is an positive integer. Our earlier established results on the integral representations for the Riemann zeta function $ζ(2n+1)$ ,$n\in\mathbb{N}$, follow directly as corollaries of these representations.

math.CA↗

arXiv2009

Derivative Polynomials and Closed-Form Higher Derivative Formulae

In a recent paper, Adamchik [V.S. Adamchik, On the Hurwitz function for rational arguments, Appl. Math. Comp. 187 (2007) 3--12] expressed in a closed form symbolic derivatives of four functions belonging to the class of functions whose derivatives are polynomials in terms of the same functions. In this sequel, simple closed-form higher derivative formulae which involve the Carlitz-Scoville higher order tangent and secant numbers are derived for eight trigonometric and hyperbolic functions.

math.CA↗

arXiv2009

A dilogarithmic integral arising in quantum field theory

Recently, an interesting dilogarithmic integral arising in quantum field theory has been closed-form evaluated in terms of the Clausen function $\text{Cl}_2(θ)$ by Coffey [J. Math. Phys.} 49 (2008), 093508]. It represents the volume of an ideal tetrahedron in hyperbolic space and is involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It is shown here, by simple and direct arguments, that this integral can be expressed by the triplet of the Clausen function values which are involved in one of the two above-mentioned conjectures.

math.CA↗