By Siegfried Bocherer, Tomoyoshi Ibukiyama, Masanobu Kaneko, Fumihiro Sato

This quantity features a invaluable number of articles provided at a convention on Automorphic types and Zeta capabilities in reminiscence of Tsuneo Arakawa, an eminent researcher in modular varieties in different variables and zeta features. The publication starts with a evaluate of his works, by way of sixteen articles by means of specialists within the fields together with H Aoki, R Berndt, ok Hashimoto, S Hayashida, Y Hironaka, H Katsurada, W Kohnen, A Krieg, A Murase, H Narita, T Oda, B Roberts, R Schmidt, R Schulze-Pillot, N Skoruppa, T Sugano, and D Zagier. quite a few themes within the idea of modular types and zeta services are lined: Theta sequence and the root difficulties, Jacobi kinds, automorphic varieties on Sp(1, q), double zeta services, specified values of zeta and L-functions, lots of that are heavily regarding Arakawa's works. This choice of papers illustrates Arakawa's contributions and the present traits in modular kinds in numerous variables and similar zeta services.

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Univ. St. Paul. 45 (1996), no. 2, 213227. 21. Koecher-Maass Dirichlet series corresponding to Jacobi forms and Cohen Eisenstein series. Comment. Math. Univ. St. Paul. 4 7 (1998), no. 1, 93-122. 22. Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153 (1999), 189-209. (with M. Kaneko) 23. On poly-Bernoulli numbers. Comment. Math. Univ. St. Paul. 48 (1999), no. 2, 159-167. (with M. Kaneko) 24. A note on the restriction map for Jacobi forms. Abh. Math. Sem.

Comment. Math. Univ. St. Paul. 4 7 (1998), no. 1, 93-122. 22. Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153 (1999), 189-209. (with M. Kaneko) 23. On poly-Bernoulli numbers. Comment. Math. Univ. St. Paul. 48 (1999), no. 2, 159-167. (with M. Kaneko) 24. A note on the restriction map for Jacobi forms. Abh. Math. Sem. Univ. Hamburg 69 (1999), 309-317. (with S. Bocherer) 25. Saito-Kurokawa lifting for odd weights. Comment. Math. Univ. St. Paul. 49 (2000), no.

A r ( 2 n + 1) From this proposition, we have the upper bounds for the dimensions of At ,A+-,Ai+and A*-. 1. In the following table, each dimension of the left hand side is not greater than the coefficient of xk on the formal power series development of the right hand side. At+ Af A~k A: + 1 2 (1 - z )(l - x 3 )(l - z 4 ) X? 2 (1 - x )(l - z 3 )(l - x 4 ) I11 (1 - x 2 )(l - x 3 )(l - x4) X5 (l-z2)(l-:c3)(l-z4) Proof. From the previous proposition, the dimension of A~£+ is not greater than oo oo oo ^dimM2fc+4„(n) = ^ d i m M 2 f c _ 8 „ = ^dimM2(fc_4n).

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