Titlebook: Number Theoretic Methods; Future Trends Shigeru Kanemitsu,Chaohua Jia Book 2002 Springer Science+Business Media Dordrecht 2002 Algebra.Arit

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,Ramanujan’s Formula and Modular Forms,ional equations, incomplete gamma series, and the like — are placed like nodes on the woofs. Some of them are woven by warps as Hecke theory or Lavrik’s theory. The former connects the modular relation to the functional equation, thus making it possible to go to and from between the more orderly wor

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Primitive Roots: A Survey,rvey a few results and conjectures on this subject, and we discuss generalizations to arbitrary moduli. A primitive root to a modulus . is a residue coprime to . which generates a cyclic subgroup of maximal order in the group of reduced residues modulo n. .

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Zeta-Functions Defined by Two Polynomials,ed by two polynomials . and ., follows. Then the holomorphy of .(.; ., .) at non-positive integers is proved, and explicit formulas for the values .(0; ., .) and .(0; ., .) are given. The latter formula gives a generalization of an explicit formula for the regularized determinant of the Laplacian on

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A Penultimate Step toward Cubic Theta-Weyl Sums,his paper we shall present basic ingredients for interpreting cubic Weyl sums as finite theta series, i.e. the cubic continued fraction expansion, the van der Corput reciprocal function, cubic reciprocal and parabolic transformations.

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Convexity and Intersection of Random Spaces,nd when . → ∞is solved rigorously. The asymptotic expression coincides with the one found by E. Gardner ([4]), using non rigorous replica calculations in neural network theory. When . is larger than .. the volume of the intersection goes to 0 more rapidly than exp(— . const). We use the cavity metho

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