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 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.