For any given totally real number field K, we compute the
Fourier developments of the Jacobi Eisenstein series over K
at the cusp at infinity. As main application we prove, for
any K with class number 1, that the L-series of the Jacobi
Eisenstein series of weight k ≥ 3 for indices with rank and
modified level 1 coincide with the L-series of the Eisenstein
series of weight 2k −2 on the full Hilbert modular group of K.
Moreover, under this correspondence the Fourier coefficients
of the Jacobi Eisenstein series are related to the twisted L-
series of the Hilbert Eisenstein series at the critical point by
a Waldspurger type identity. This is a first step in the proof
that Skoruppa’s and Zagier’s lifting from Jacobi forms over
Q to elliptic modular forms holds true over arbitrary totally
real number fields too.