On simultaneous diophantine approximations to ζ(2)\zeta(2) and ζ(3)\zeta(3)

We present a hypergeometric construction of rational approximations to $\zeta(2)$ and $\zeta(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, $\zeta(2)$ and $\zeta(3)$ with rational coefficients. A new notion of (simultaneous) diophantine exponent is introduced to formalise the arithmetic structure of these specific linear forms. Finally, the properties of this newer concept are studied and linked to the classical irrationality exponent and its generalisations given recently by S. Fischler.Comment: 23 pages; v2: new subsection 4.5 adde

On Recurrences for Ising Integrals

We use WZ-summation methods to compute recurrences for the Ising-class integrals Cn,k. In this context, we describe an algorithmic approach to obtain homogeneous and inhomogeneous recurrences for a general class of multiple contour integrals of Barnes ’ type.

Experimental mathematics and mathematical physics

One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory

The elliptic dilogarithm for the sunset graph

We study the sunset graph defined as the scalar two-point self-energy at two-loop order. We evaluate the sunset integral for all identical internal masses in two dimensions. We give two calculations for the sunset amplitude; one based on an interpretation of the amplitude as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the amplitude in this case is a family of periods associated to the universal family of elliptic curves over the modular curve X_1(6). We show that the integral is given by an elliptic dilogarithm evaluated at a sixth root of unity modulo periods. We explain as well how this elliptic dilogarithm value is related to the regulator of a class in the motivic cohomology of the universal elliptic family.Comment: 3 figures, 43 pages. v2: minor corrections. version to be published in The Journal of Number Theor

Q\mathbb Q-linear dependence of certain Bessel moments

Let $I_0$ and $K_0$ be modified Bessel functions of the zeroth order. We use Vanhove's differential operators for Feynman integrals to derive upper bounds for dimensions of the $\mathbb Q$-vector space spanned by certain sequences of Bessel moments $$\left\{\left.\int_0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm{d}\, t\right|k\in\mathbb Z_{\geq0}\right\},$$where $a$ and $b$ are fixed non-negative integers. For $a\in\mathbb Z\cap[1,b)$, our upper bound for the $\mathbb Q$-linear dimension is $\lfloor (a+b-1)/2\rfloor$, which improves the Borwein-Salvy bound $\lfloor (a+b+1)/2\rfloor$. Our new upper bound $\lfloor (a+b-1)/2\rfloor$ is not sharp for $a=2,b=6$, due to an exceptional $\mathbb Q$-linear relation $\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t\mathrm{d}\, t=72\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t^{3}\mathrm{d}\, t$, which is provable by integrating modular forms.Comment: (v1) i+21 pages. Simplification and extension of some results in Section 5 of arXiv:1706.08308; (v2) i+29 pages; (v3) 20 pages, accepted versio