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 of 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 of 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 of level N and then use it to solve a complicated under-determined
system of linear equations by some standard techniques from linear algebra.Comment: A new section 5 is added and a new Theorem 6.5 is prove
