Elementary Dirichlet series and modular forms

Goro Shimura9780387724737, 0387724737

The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms.  As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method.  After a review of elementry Fourier analysis, we present completely new results with new methods, though old results will also be proved.  No advanced knowledge of number theory is required up to this point.  As applications, new formulas for the second factor of the class number of a cyclotomic field will be given.  The second half of the book assumes familiarity with basic knowledge of modular forms.  However, all definitions and facts are clearly stated, and precise references are to the determination of the critical values of Hecke L-functions of an imaginary quadratic field.  Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of ellitic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight.