Values of Zeta Functions and Their Applications

Zagier, Don

doi:10.1007/978-3-0348-9112-7_23

Cited by 550 publications

(588 citation statements)

References 0 publications

Supporting

Mentioning

566

Contrasting

Unclassified

Order By: Relevance

“…+ a 0 = 0 (110) Remark 4. In this work, neither the study of dim Z p [50] (see also [8], conjecture C2) nor the estimate of the number of A-irreducible polyzêtas generating Z p , are discussed knowing the A-irreducible polyzêtas form transcendence basis of the A-algebra Z.…”

Section: Structure and Arithmetical Nature Of Polyzêtasmentioning

confidence: 99%

On a conjecture by Pierre Cartier about a group of associators

Minh

2013

Acta Math Vietnam.

View full text Add to dashboard Cite

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

show abstract

Section: Structure and Arithmetical Nature Of Polyzêtasmentioning

confidence: 99%

On a conjecture by Pierre Cartier about a group of associators

Minh

2013

Acta Math Vietnam.

View full text Add to dashboard Cite

show abstract

“…The first integral I 1 involves the Beta function (21) and the hypergeometric function 3 F 2 (22). In fact, the latter integral is f 2 of Eq.…”

Section: F 2 : Triple Hypergeometric Function With a Single Polementioning

confidence: 99%

Multigluon scattering in open superstring theory

2006

View full text Add to dashboard Cite

We discuss the amplitudes describing N -gluon scattering in type I superstring theory, on a disk world-sheet. After reviewing the general structure of amplitudes and the complications created by the presence of a large number of vertices at the boundary, we focus on the most promising case of maximally helicity violating (MHV) configurations because in this case, the zero Regge slope limit (α ′ → 0) is particularly simple. We obtain the full-fledged MHV disk amplitudes for N = 4, 5 and N = 6 gluons, expressed in terms of one, two and six functions of kinematic invariants, respectively. These functions represent certain boundary integrals -generalized Euler integrals -which for N ≥ 6 correspond to multiple hypergeometric series (generalized Kampé de Fériet functions). Their α ′ -expansions lead to Euler-Zagier sums. For arbitrary N , we show that the leading string corrections to the Yang-Mills amplitude, of order O(α ′2 ), originate from the well-known α ′2 TrF 4 effective interactions of four gauge field strength tensors. By using iteration based on the soft gluon limit, we derive a simple formula valid to that order for arbitrary N . We argue that such a procedure can be extended to all orders in α ′ . If nature gracefully picked a sufficiently low string mass scale, our results would be important for studying string effects in multi-jet production at the Large Hadron Collider (LHC).

show abstract

“…Our series development starts with the observation that ζ admits of certain integral representations (see Crandall [10], Borwein et al [7], and Zagier [14]). Denote the argument vector s = {s 1 , s 2 , ..., s d }, and consider a particular d-dimensional integral representation:…”

Section: Integral Representationmentioning

confidence: 99%

“…The multiple zeta sums: [3], Borwein et al [4], Borwein et al [7], Broadhurst et al [9], Crandall and Buhler [11], Markett [13] and Zagier [14]) (note that some previous treatments have reversed ordering of the indices n i ). The study of such sums is not only important to general zeta function theory, but also touches upon such domains as knot theory and particle physics methodology.…”

Section: Introductionmentioning

confidence: 99%

Fast evaluation of multiple zeta sums

Crandall

1998

Math. Comp.

View full text Add to dashboard Cite

Abstract. We show that the multiple zeta sum:for positive integers s i with s 1 > 1, can always be written as a finite sum of products of rapidly convergent series. Perhaps surprisingly, one may develop fast summation algorithms of such efficiency that the overall complexity can be brought down essentially to that of one-dimensional summation. In particular, for any dimension d one may resolve D good digits of ζ in O(D log D/ log log D) arithmetic operations, with the implied big-O constant depending only on the set {s 1 , ..., s d }.

show abstract

Values of Zeta Functions and Their Applications

Cited by 550 publications

References 0 publications

On a conjecture by Pierre Cartier about a group of associators

On a conjecture by Pierre Cartier about a group of associators

Multigluon scattering in open superstring theory

Fast evaluation of multiple zeta sums

Contact Info

Product

Resources

About