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

@article{Cvijovic2010ProofOT, title={Proof of the Borwein-Broadhurst conjecture for a dilogarithmic integral arising in quantum field theory}, author={Djurdje Cvijovic}, journal={arXiv: Mathematical Physics}, year={2010} }

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…

## References

SHOWING 1-10 OF 23 REFERENCES

### Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links

- Mathematics
- 1998

We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $$\frac{a}{b}\,{\rm vol}({\cal…

### Solving differential equations for 3-loop diagrams: relation to hyperbolic geometry and knot theory

- Mathematics
- 1998

In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3 dimensions was found, using the lattice algorithm PSLQ. Here we give a more general formula, involving 3 distinct masses.…

### Massive 3-loop Feynman diagrams reducible to SC$^*$ primitives of algebras of the sixth root of unity

- Mathematics
- 1999

Abstract. In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the…

### Elliptic integral evaluations of Bessel moments and applications

- Mathematics
- 2008

We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical…

### Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

- Mathematics
- 2008

We calculate very long low- and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ(5) and six-particle…

### Evaluation of a ln tan integral arising in quantum field theory

- Mathematics
- 2008

We analytically evaluate a dilogarithmic integral that is prototypical of volumes of ideal tetrahedra in hyperbolic geometry. We additionally obtain new representations of the Clausen function Cl2…

### The evaluation of character Euler double sums

- Mathematics
- 2008

Euler considered sums of the form $$\sum_{m=1}^{\infty}\frac{1}{m^{s}}\sum_{n=1}^{m-1}\frac{1}{n^{t}}.$$ Here natural generalizations of these sums namely…

### Highly Parallel, High-Precision Numerical Integration

- Mathematics, Computer Science
- 2005

It is believed that this and one other 20,000-digit integral evaluation that are reported are the highest-precision non-trivial numerical integrations performed to date.

### Elementary Number Theory

- Mathematics
- 1976

Some preliminary considerations divisibility theory in the integers primes and their distribution the theory of congruences Fermat's theorem number-theoretic functions Euler's generalization of…

### Ten Problems in Experimental Mathematics

- MathematicsAm. Math. Mon.
- 2006

The intent was to present ten problems that are characteristic of the sorts of problems that commonly arise in ''experimental mathematics'' and to present a concise account of how one combines symbolic and numeric computation, which may be termed ''hybrid computation'', in the process of mathematical discovery.