Double zeta values, double Eisenstein series, and modular forms of level $$2$$

Double shuffle relations of double zeta values and the double Eisenstein series at level N

In their seminal paper, 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 ...

Infinite Series Forms of Double Integrals

where 1 2 1 2 , , , r r θ θ are any real numbers, and n is any positive integer. We can obtain the infinite series forms of these double integrals using Taylor series expansions and integration term by term theorem; these are the major results of this paper (i.e., Theorems 1-3). Adams et al. [1], Nyblom [2], and Oster [3] provided some techniques to solve the integral problems. Yu [4-29], Yu an...

The Eisenstein cocycle, partial zeta values and Gross–Stark units

We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the “order of vanishing” of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross–Stark units.

Some Notes on Weighted Sum Formulae for Double Zeta Values

We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta values, and the Witten zeta function. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums fr...

Moments of the Riemann Zeta Function and Eisenstein Series I