Values of zeta functions and their applications.

*(English)*Zbl 0822.11001
Joseph, A. (ed.) et al., First European congress of mathematics (ECM), Paris, France, July 6-10, 1992. Volume II: Invited lectures (Part 2). Basel: Birkhäuser. Prog. Math. 120, 497-512 (1994).

According to the author, “the purpose of this article is neither to prove new results nor to give a survey of a well-defined area of mathematics”, rather this paper aims “to give a feel of some of the ways in which special values of zeta-functions interrelate with other interesting mathematical questions”.

The author starts with a simple proof of rationality of the numbers \(\pi^{-2k} \zeta (2k)\), \(k>0\), \(k\in \mathbb Z\), for the Riemann zeta-function \(\zeta(s)\), and, after a brief description of the general conjectures relating to the critical values of motivic \(L\)-functions, proceeds to comment on: the recent status of the Birch and Swinnerton-Dyer conjecture; the periods of modular forms; the polylogarithms and algebraic \(K\)-theory; his recent work with F. Rodriguez-Villegas (cf. F. Rodriguez-Villegas and D. Zagier in [Gouvêa F. Q. (ed.), Advances in number theory, 81–99 (1993; Zbl 0791.11060)]) concerning the special value of a Hecke \(L\)-series of an imaginary quadratic field at the point of symmetry of the functional equation; the invariants of moduli spaces in relation to the special values of classical zeta-functions as well as of some new zeta-functions arising in mathematical physics (Verlinde formulae). He goes on to define “multiple zeta-values” as sums \(\sideset\and {^*}\to\sum_ n \prod^ r_{i=1} n_ i^{-k_ i}\), \(k_ i\geq 1\), \(k_ r\geq 2\), with \(\sideset\and {^*}\to\sum\) ranging over all the integers under condition \(n_ r> \cdots> n_ 1> 0\), and points out that the graded ring of these zeta-values is related to the category of mixed Tate motives, to the values of the Drinfeld (multiple) integrals, and to the structure of the set of Vassiliev knot invariants.

The last chapter of this survey provides an annotated list of references for further reading.

For the entire collection see [Zbl 0807.00008].

The author starts with a simple proof of rationality of the numbers \(\pi^{-2k} \zeta (2k)\), \(k>0\), \(k\in \mathbb Z\), for the Riemann zeta-function \(\zeta(s)\), and, after a brief description of the general conjectures relating to the critical values of motivic \(L\)-functions, proceeds to comment on: the recent status of the Birch and Swinnerton-Dyer conjecture; the periods of modular forms; the polylogarithms and algebraic \(K\)-theory; his recent work with F. Rodriguez-Villegas (cf. F. Rodriguez-Villegas and D. Zagier in [Gouvêa F. Q. (ed.), Advances in number theory, 81–99 (1993; Zbl 0791.11060)]) concerning the special value of a Hecke \(L\)-series of an imaginary quadratic field at the point of symmetry of the functional equation; the invariants of moduli spaces in relation to the special values of classical zeta-functions as well as of some new zeta-functions arising in mathematical physics (Verlinde formulae). He goes on to define “multiple zeta-values” as sums \(\sideset\and {^*}\to\sum_ n \prod^ r_{i=1} n_ i^{-k_ i}\), \(k_ i\geq 1\), \(k_ r\geq 2\), with \(\sideset\and {^*}\to\sum\) ranging over all the integers under condition \(n_ r> \cdots> n_ 1> 0\), and points out that the graded ring of these zeta-values is related to the category of mixed Tate motives, to the values of the Drinfeld (multiple) integrals, and to the structure of the set of Vassiliev knot invariants.

The last chapter of this survey provides an annotated list of references for further reading.

For the entire collection see [Zbl 0807.00008].

Reviewer: B. Z. Moroz (Bonn)

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11M32 | Multiple Dirichlet series and zeta functions and multizeta values |

11G55 | Polylogarithms and relations with \(K\)-theory |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11R42 | Zeta functions and \(L\)-functions of number fields |

11R70 | \(K\)-theory of global fields |

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

14H60 | Vector bundles on curves and their moduli |