Papers by Henrik Bachmann
arXiv (Cornell University), Sep 16, 2013
We study the algebra MD of generating function for multiple divisor sums and its connections to m... more We study the algebra MD of generating function for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD. The (quasi-)modular forms for the full modular group SL 2 (Z) constitute a subalgebra of MD this also yields linear relations in MD. Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those in length 2, coming from modular forms. Contents 1 Introduction 2 The algebra of generating function of multiple divisor sums 3 A derivation and linear relations in MD 4 The subalgebra of (quasi-)modular forms 5 Experiments and conjectures: dimensions 6 Interpretation as a q-analogue of multiple zeta values 7 Applications to multiple zeta values References
arXiv (Cornell University), Aug 24, 2017
We study a class of q-analogues of multiple zeta values given by certain formal q-series with rat... more We study a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we present dimension conjectures for the spaces of their weight-and depth-graded parts, which have a similar shape as the conjectures of Zagier and Broadhurst-Kreimer for multiple zeta values.
arXiv (Cornell University), Apr 23, 2017
This work is an example driven overview article of recent works on the connection of multiple zet... more This work is an example driven overview article of recent works on the connection of multiple zeta values, modular forms and q-analogues of multiple zeta values given by multiple Eisenstein series. Contents 1 Multiple Eisenstein series 14 2 Multiple divisor-sums and their generating functions 24 3 Bi-brackets and a second product expression for brackets 33 4 Regularizations of multiple Eisenstein series 42 5 q-analogues of multiple zeta values 56 References 63
arXiv (Cornell University), Aug 29, 2018
We present explicit formulas for Hecke eigenforms as linear combinations of qanalogues of modifie... more We present explicit formulas for Hecke eigenforms as linear combinations of qanalogues of modified double zeta values. As an application, we obtain period polynomial relations and sum formulas for these modified double zeta values. These relations have similar shapes as the period polynomial relations of Gangl, Kaneko and Zagier and the usual sum formulas for classical double zeta values.
arXiv (Cornell University), Jul 1, 2018
We study special values of finite multiple harmonic q-series at roots of unity. These objects wer... more We study special values of finite multiple harmonic q-series at roots of unity. These objects were recently introduced by the authors and it was shown that they have connections to finite and symmetric multiple zeta values and the Kaneko-Zagier conjecture. In this note we give new explicit evaluations for finite multiple harmonic q-series at roots of unity and prove Ohno-Zagier-type relations for them.
Modular program system KAPROS for efficient management of complex reactor calculations
arXiv (Cornell University), Jan 14, 2015
We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof o... more We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov coproduct on Hopf algebras of iterated integrals.
Research in the Mathematical Sciences, Jul 25, 2023
We construct a family of q-series with rational coefficients satisfying a variant of the extended... more We construct a family of q-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given Q-valued solution of the extended double shuffle equations. We call these q-series combinatorial (bi-)multiple Eisenstein series, and in depth one they coincide with (classical) Eisenstein series. Combinatorial multiple Eisenstein series can be seen as an interpolation between the given Q-valued solution of the extended double shuffle equations (as q → 0) and multiple zeta values (as q → 1). In particular, they are q-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould.
arXiv (Cornell University), Nov 13, 2017
We give explicit formulas for the recently introduced Schur multiple zeta values, which generaliz... more We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.
arXiv (Cornell University), Aug 14, 2019
We present new determinant expressions for regularized Schur multiple zeta values. These generali... more We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.
arXiv (Cornell University), Dec 20, 2022
Multiple Eisenstein series are holomorphic functions in the complex upperhalf plane, which can be... more Multiple Eisenstein series are holomorphic functions in the complex upperhalf plane, which can be seen as a crossbreed between multiple zeta values and classical Eisenstein series. They were originally defined by Gangl-Kaneko-Zagier in 2006, and since then, many variants and regularizations of them have been studied. They give a natural bridge between the world of modular forms and multiple zeta values. In this note, we give a new algebraic interpretation of stuffle regularized multiple Eisenstein series based on the Hopf algebra structure of the harmonic algebra introduced by Hoffman.
arXiv (Cornell University), Jan 16, 2018
Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author in... more Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the noncommutative polynomial algebra in two letters. These give a class of relations among multiple zeta values, which are known to be a subclass of the socalled linear part of the Kawashima relations. In this paper we show the opposite implication, that is the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.
arXiv (Cornell University), Sep 16, 2013
We study the algebra MD of multiple divisor functions and its connections to multiple zeta values... more We study the algebra MD of multiple divisor functions and its connections to multiple zeta values. Multiple divisor functions are formal power series in q with coefficients in Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD. The (quasi-)modular forms for the full modular group SL_2(Z) constitute a sub-algebra of MD this also yields linear relations in MD. Multiple divisor functions can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those in length 2, coming from modular forms.
arXiv (Cornell University), Apr 30, 2015
We study the algebra of certain q-series, called bi-brackets, whose coefficients are given by wei... more We study the algebra of certain q-series, called bi-brackets, whose coefficients are given by weighted sums over partitions. These series incorporate the theory of modular forms for the full modular group as well as the theory of multiple zeta values (MZV) due to their appearance in the Fourier expansion of regularised multiple Eisenstein series. Using the conjugation of partitions we obtain linear relations between bi-brackets, called the partition relations, which yield naturally two different ways of expressing the product of two bi-brackets similar to the stuffle and shuffle product of multiple zeta values. Bi-brackets are generalizations of the generating functions of multiple divisor sums, called brackets, [s1, . . . , s l ] studied in [BK]. We use the algebraic structure of bi-brackets to define further q-series [s1, . . . , s l ] ¡ and [s1, . . . , s l ] * which satisfy the shuffle and stuffle product formulas of MZV by using results about quasi-shuffle algebras introduced by Hoffman. In [BT] regularised multiple Eisenstein series G ¡ were defined, by using an explicit connection to the coproduct on formal iterated integrals. These satisfy the shuffle product formula. Applying the same concept for the coproduct on quasi-shuffle algebras enables us to define multiple Eisenstein series G * satisfying the stuffle product. We show that both G ¡ and G * are given by linear combinations of products of MZV and bi-brackets. Comparing these two regularized multiple Eisenstein series enables us to obtain finite double shuffle relations for multiple Eisenstein series in low weights which extend the relations proven in [BT].
arXiv (Cornell University), Dec 6, 2023
We introduce the algebra of formal multiple Eisenstein series and study its derivations. This alg... more We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an sl 2 -algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms.
In [Ok] Okounkov studies a specific q-analogue of multiple zeta values and makes some conjectures... more In [Ok] Okounkov studies a specific q-analogue of multiple zeta values and makes some conjectures on their algebraic structure. In this note we compare Okounkovs q-analogues to the generating function for multiple divisor sums defined in [BK1]. We also state a conjecture on their dimensions that complements Okounkovs conjectural formula and present some numerical evidences for it. 2 q-analogues of multiple zeta values
Sum formulas for Schur multiple zeta values
Journal of Combinatorial Theory, Series A
Sum formulas for Schur multiple zeta values
arXiv (Cornell University), Feb 6, 2023
Cornell University - arXiv, Jul 25, 2014
In [Ok] Okounkov studies a specific q-analogue of multiple zeta values and makes some conjectures... more In [Ok] Okounkov studies a specific q-analogue of multiple zeta values and makes some conjectures on their algebraic structure. In this note we compare Okounkovs q-analogues to the generating function for multiple divisor sums defined in [BK1]. We also state a conjecture on their dimensions that complements Okounkovs conjectural formula and present some numerical evidences for it. 2 q-analogues of multiple zeta values
Cornell University - arXiv, Sep 9, 2021
We introduce the formal double Eisenstein space E k , which is a generalization of the formal dou... more We introduce the formal double Eisenstein space E k , which is a generalization of the formal double zeta space D k of Gangl-Kaneko-Zagier, and prove analogues of the sum formula and parity result for formal double Eisenstein series. We show that Q-linear maps E k → A, for some Q-algebra A, can be constructed from formal Laurent series (with coefficients in A) that satisfy the Fay identity. As the prototypical example, we define the Kronecker realization ρ K : E k → Q[[q]], which lifts Gangl-Kaneko-Zagier's Bernoulli realization ρ B : D k → Q, and whose image consists of quasimodular forms for the full modular group. As an application to the theory of modular forms, we obtain a purely combinatorial proof of Ramanujan's differential equations for classical Eisenstein series.
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Papers by Henrik Bachmann