Special values ζQ(k) (k = 2, 3, 4, ...) of the spectral zeta function ζQ(s) of the non-commutative harmonic oscillator Q are discussed. Particular emphasis is put on basic modular properties of the generating function w k (t) of Apéry-like numbers which is appeared in analysis on the first anomaly of each special value. Here the first anomaly is defined to be the "1st order" difference of ζQ(k) from ζ(k), ζ(s) being the Riemann zeta function. In order to describe such modular properties for k ≥ 4, we introduce a notion of residual modular forms for congruence subgroups of SL2(Z) which contains the classical notion of Eichler integrals as a particular case. Further, we define differential Eisenstein series, which are residual modular forms. Using such differential Eisenstein series, for example, one obtains an explicit description of w4(t). A certain Eichler cohomology group associated to such residual modular forms plays also an important role in the discussion.
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