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Abstract. Special values ζ_Q(k) (k = 2, 3, 4, ...) of the spectral zeta function ζ_Q(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 Apery-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 ζ_Q(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 SL_2(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 w_4(t). A certain Eichler cohomology group associated to such residual modular forms plays also an important role in the discussion.

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