Renormalization of multiple zeta values

By Li Guo, Bin Zhang. 2006

142 citations recorded by the source corpus.

Abstract Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually undefined. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara–Kaneko–Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/6adb6894acfa43c458fd38aaa96bdf369f3c6811

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

On integro-differential algebras (2012, background): “Rota-Baxter algebras have found extensive applications in mathematics and physics, including quantum field theory and the classical Yang-Baxter equation [4, 12, 13, 16, 17, 23].”
Nested sums of symbols and renormalized multiple zeta values (2010, methodology): “Both authors are indebted to Li Guo for various interesting discussions and for bringing to our attention and commenting for us his own work with Bin Zhang [ GZ ] which also uses a Birkhoff factorisation to renormalise multiple zeta functions.”
Operated semigroups, Motzkin paths and rooted trees (2007, methodology): “Baxter operator of weight −1 [ 31 ] that commute with each other.”
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, background): “Recently, there have been several interesting developments of Rota–Baxter algebras in theoretical physics and mathematics, including quantum field theory [12, 13, 43, 44, 51], associative Yang–Baxter equations [1, 2], shuffle products [18, 36, 37], operads [4, 15, 19, 45, 46], Hopf algebras [6, 18],”
Rota–Baxter algebras and dendriform algebras (2005, background): “ including quantum ﬁeld theory [14, 15], Yang-Baxter equations [1, 2, 3], shuﬄe products [19, 41, 42], operads [4, 16, 20, 50, 51, 52], Hopf algebras [8, 19, 31], combinatorics [38] and number theory [19, 23, 39, 45, 60, 61, 71]. The most prominent of these is the work [14, 15] of Connes and Kreimer”
Euler's constant: Euler's work and modern developments (2013, background): “Zhang [150], [151], Guo, Paycha, Xie, and Zhang [149], and Manchon and Paycha [211].”
Splitting of operations, Manin products and Rota-Baxter operators (2011, background): “Rota [37] and found many app lications in the last decade in mathematics and physics [1, 4, 5, 15, 18, 19, 39], especially in the Connes-Kreimer approach of renormalization in quantum field theory [10, 16, 20, 35].”
On Differential Rota-Baxter Algebras (2007, background): “morphisms. It is proved in [18] that X(A) with the mixable shuﬄe product is the free commutative Rota-Baxter algebra on A. The mixable shuﬄe product is shown to be the same as the quasi-shuﬄe product [11, 20, 21]. Since ⋄ is compatible with the multiplication in A, we will often suppress the symbol ”