Academia.edu no longer supports Internet Explorer.
To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser.
2006, Automorphic Forms and Zeta Functions - Proceedings of the Conference in Memory of Tsuneo Arakawa
AI
This paper explores the properties of double zeta values and their deep connections to modular forms. Key findings include new relations among double zeta values, their representation as formal symbols, and the establishment of a Q-vector space structure related to modular forms of weight k. Additionally, the introduction of double Eisenstein series provides fresh insights into these relations, highlighting their role in algebraic representations of double zeta values.
Journal of the London Mathematical Society, 2015
In their seminal paper "Double zeta values and modular forms" Gangl, Kaneko and Zagier defined a double Eisenstein series and used it to study the relations between double zeta values. One of their key ideas is to study the formal double space and apply the double shuffle relations. They also proved the double shuffle relations for the double Eisenstein series. More recently, Kaneko and Tasaka extended the double Eisenstein series to level 2, proved its double shuffle relations and studied the double zeta values at level 2. Motivated by the above works, we define in this paper the corresponding objects at higher levels and prove that the double Eisenstein series at level N satisfies the double shuffle relations for every positive integer N. In order to obtain our main theorem we prove a key result on the multiple divisor functions at level N and then use it to solve a complicated under-determined system of linear equations by some standard techniques from linear algebra. Contents 1. Introduction 1 2. The multiple zeta values at level N 3 3. Double zeta space at level N 4 4. Fourier series expansion of the double Eisenstein series at level N 9 5. Decomposition of the zeta values at level N 12 6. Double shuffle relations of double Eisenstein series at level N 15 7. A key relation on multiple divisor functions at level N 20 8. A linear algebra problem 23 References 30
京都大学数理解析研究所, 2017
We study a certain class of q-analogues of multiple zeta values, which appear in the Fourier expansion of multiple Eisenstein series. Studying their algebraic structure and their derivatives we propose conjectured explicit formulas for the derivatives of double and triple Eisenstein series.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2020
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.
2021
In this survey article, we discuss the algebraic structure of q-analogues of multiple zeta values, which are closely related to derivatives of Eisenstein series. Moreover, we introduce the formal double Eisenstein space, which generalizes the formal double zeta space of Gangl, Kaneko, and Zagier. Using the algebraic structure of q-analogues of multiple zeta values, we will present a realization of this space. As an application, we will obtain purely combinatorial proofs of identities among (quasi-)modular forms.
2016
In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of 'zeta element', of 'Weil-étale cohomology complexes' and of 'integral Selmer groups' for the multiplicative group Gm over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fields. We also show that the conjecture entails much detailed information about the arithmetic properties of generalized Stark elements including a new family of integral congruence relations between Rubin-Stark elements (that refines recent conjectures of Mazur and Rubin and of the third author) and explicit formulas in terms of these elements for the higher Fitting ideals of the integral Selmer groups of Gm, thereby obtaining a clear and very general approach to the theory of abelian Stark conjectures. As first applications of this approach, we derive, amongst other things, a proof of (a refinement of) a conjecture of Darmon concerning cyclotomic units, a proof of (a refinement of) Gross's 'Conjecture for Tori' in the case that the base field is Q, a proof of new cases of the equivariant Tamagawa number conjecture in situations in which the relevant p-adic L-functions have trivial zeroes, explicit conjectural formulas for both annihilating elements and, in certain cases, the higher Fitting ideals (and hence explicit structures) of ideal class groups, a reinterpretation of the p-adic Gross-Stark Conjecture in terms of the properties of zeta elements and a strong refinement of many previous results (of several authors) concerning abelian Stark conjectures.
2010
We present nine lectures that are introductory and foundational in nature. The basic inspiration comes from the Riemann zeta function, which is the starting point. Along the way there are sprinkled some connections of the material to physics. The asymptotics of Fourier coefficients of zero weight modular forms, for example, are considered in regards to black hole entropy. Thus we have some interests also connected with Einstein’s general relativity. References are listed that cover much more material, of course, than what is attempted here. Although his papers were few in number during his brief life, which was cut short by tuberculosis, Georg Friedrich Bernhard Riemann (1826–1866) ranks prominently among the most outstanding mathematicians of the nineteenth century. In particular, Riemann published only one paper on number theory [32]: “Uber die Anzahl der Primzahlen unter einer gegebenen Grosse”, that is, “On the number of primes less than a given magnitude”. In this short paper p...
Cornell University - arXiv, 2014
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
The Ramanujan Journal, 2013
The even weight period polynomial relations in the double shuffle Lie algebra ds were discovered by Ihara, and completely classified in [Sch06] by relating them to restricted even period polynomials associated to cusp forms on SL 2 (Z). In an article published in the same year, Gangl, Kaneko and Zagier [GKZ06] displayed certain linear combinations of odd-component double zeta values which are equal to scalar multiples of simple zeta values in even weight, and also related them to restricted even period polynomials. In this paper, we relate the two sets of relations, showing how they can be deduced from each other by duality. † Let y i = x i-1 y, so that all words ending in y can be written as y i 1 • • • y ir . The stuffle product of two such words is defined recursively by 1 * w = w * 1 = w and y i w * y j w ′ = y i (w * y j w ′ ) + y j (y i w * w ′ ) + y i+j (w * w ′ ).
We expose and exploit relationship between Riemann and Hurwitz zeta functions basically to study sums of the form $\frac{1}{\pi ^{n}}% \sum_{j=-\infty }^{\infty }1/(jk+l)^{n}$ and $\frac{1}{\pi ^{n}}% \sum_{j=-\infty }^{\infty }(-1)^{j}/(jk+l)^{n}$ for integer $n,k>1,$ and $% l\leq k/2$. We show that for odd $k$ or $k\allowbreak =\allowbreak 2^{m}$ for some $m$ natural first sums while for even $k$ second sums are algebraic numbers. Hence we generalize known result for $k\allowbreak =\allowbreak 4$ and $l\allowbreak =\allowbreak 1$ dating back to L. Euler. Besides we recall and prove in believed to be new way known relationships between Hurwitz and Riemann Zeta functions.
Journal of the Mathematical Society of Japan, 2020
We introduce finite and symmetric Mordell-Tornheim type of multiple zeta values and give a new approach to the Kaneko-Zagier conjecture stating that the finite and symmetric multiple zeta values satisfy the same relations. 1991 Mathematics Subject Classification. 11M32, 11R18, 05A30. Key words and phrases. Kaneko-Zagier conjecture (finite multiple zeta value and symmetric multiple zeta value), Mordell-Tornheim multiple zeta values, Mordell-Tornheim-Witten (q-)multiple zeta values.
Our main aim in this series of articles is to present a clear new view, generalization and refinement of a range of well-known results and conjectures concerning the arithmetic properties of zeta elements. In this first article we study the L-functions that are attached to the multiplicative group over a finite abelian extension of global fields.
arXiv (Cornell University), 2013
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: Number Theory, 2016
We study a certain class of q-analogues of multiple zeta values, which appear in the Fourier expansion of multiple Eisenstein series. Studying their algebraic structure and their derivatives we propose conjectured explicit formulas for the derivatives of double and triple Eisenstein series.
Journal of Algebra, 2003
We establish a new class of relations among the multiple zeta values ζ(k 1 , . . . , k l ) = n 1 >···>n l ≥1
Springer Proceedings in Mathematics & Statistics, 2020
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:1704.06930v1 [math.NT] 23 Apr 2017 i.e. up to the lower weight term − 1 12 [3] and 1 12 [2, 2] − 1 2 [2, 3] − 1 12 [3, 1] this looks exactly like (1),(2). One might ask if there is also something which corresponds to the shuffle product (3) of multiple zeta values. It turned out that for the lowest length case, this has to do with the differential operator d = q d dq. In [BK] it was shown that [2] • [3] = [2, 3] + 3[3, 2] + 6[4, 1] − 3[4] + d[3] , (7) which, again up to the term −3[4] + d[3], looks exactly like the shuffle product (3) of multiple zeta values. In particular it follows that d[3] is again in the space MD and in general it was shown that Theorem 4. ([BK]) The operator d = q d dq is a derivation on MD. ii) Equation (7) above was the motivation to study a larger class of q-series, which will be called bi-brackets. While the quasi-shuffle product of brackets also exists in higher length, the second expression for the product, corresponding to the shuffle product, does not appear in higher length if one just allows derivatives as "error terms". The bibrackets can be seen as a generalization of the derivative of brackets. For
The Ramanujan Journal, 2015
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 2 The algebra of generating function of multiple divisor sums 9 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 Theorem 1.6. We have the following exact values or lower bounds for dim Q Fil W,L k,l (MD) Theorem 1.7. The operator d = q d dq is a derivation on MD, it maps Fil W,L k,l (MD) to Fil W,L k+2,l+1 (MD). [8] = 1 40 [4] − 1 252 [2] + 12[4, 4] .
Advances in Mathematics, 2021
We examine an unstudied manuscript of N. S. Koshliakov over 150 pages long and containing the theory of two interesting generalizations ζp(s) and ηp(s) of the Riemann zeta function ζ(s), which we call Koshliakov zeta functions. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each p ∈ R +. The first one is a generalization of Ramanujan's famous formula for ζ(2m + 1) and the second is an elegant extension of a modular relation on page 220 of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for ζ(4m + 3).
2017
We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the analytic continuations of these “partition zeta functions,” find unusual formulas for the Riemann zeta function, prove identities for multiple zeta values, and see that some of the formulas allow for p-adic interpolation. The second family we study was anticipated by Manin and makes use of modular forms, functions which are intimately related to integer partitions by universal polynomial recurrence relations. We survey recent work on these zeta polynomials, including the proof of their Riemann Hypothesis.
Contemporary Mathematics, 2015
Let F denote the free polynomial algebra F = Q s 3 , s 5 , s 7 ,. .. on non-commutative variables s i for odd i ≥ 3. The algebra F is weight-graded by letting sn be of weight n; we write Fn for the weight n part. In this paper we put a "special" decreasing depth filtration F = F 1 ⊃ F 2 ⊃ • • • ⊃ F d ⊃ F d+1 • • • on F , based on the period polynomials associated to cusp forms on SL 2 (Z). We define a lattice L of particular combinatorially defined subspaces of F , and conjecture that this lattice is distributive. Assuming this conjecture, we show that the dimensions of the weight n filtered quotients F d n /F d+1 n are given by the coefficients of the well-known Broadhurst-Kreimer generating series, defined by them to predict dimensions for the algebra of multiple zeta values. We end by explaining the expected relationship between F equipped with the special depth filtration and the algebras of formal and motivic multiple zeta values.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.