Topics in Analytic Number Theory

At the time of Professor Rademacher's death early in 1969, there was available a complete manuscript of the present work. The editors had only to supply a few bibliographical references and to correct a few misprints and errors. No substantive changes were made in the manu script except in one or two places where references to additional material appeared; since this material was not found in Rademacher's papers, these references were deleted. The editors are grateful to Springer-Verlag for their helpfulness and courtesy. Rademacher started work on the present volume no later than 1944; he was still working on it at the inception of his final illness. It represents the parts of analytic number theory that were of greatest interest to him. The editors, his students, offer this work as homage to the memory of a great man to whom they, in common with all number theorists, owe a deep and lasting debt. E. Grosswald Temple University, Philadelphia, PA 19122, U.S.A. J. Lehner University of Pittsburgh, Pittsburgh, PA 15213 and National Bureau of Standards, Washington, DC 20234, U.S.A. M. Newman National Bureau of Standards, Washington, DC 20234, U.S.A. Contents I. Analytic tools Chapter 1. Bernoulli polynomials and Bernoulli numbers ....... . 1 1. The binomial coefficients ..................................... . 1 2. The Bernoulli polynomials .................................... . 4 3. Zeros of the Bernoulli polynomials ............................. . 7 4. The Bernoulli numbers ....................................... . 9 5. The von Staudt-Clausen theorem .............................. . 10 6. A multiplication formula for the Bernoulli polynomials ........... .

Inhalt
I. Analytic tools.- 1. Bernoulli polynomials and Bernoulli numbers.- 2. The Euler-MacLaurin sum formula.- 3. The ?-function and Mellin's theorem.- 4. The Phragmén-Lindelöf theorem.- 5. The Poisson sum formula and applications.- II. Special functions.- 6. The Riemann ?-function.- 7. About the prime-number theorem and the zeros of the ?-function.- 8. The Eisenstein series.- 9. The transformation of log ?(?) and the theory of the Dedekind sums.- 10. The ?- functions.- 11. Elliptic functions and their applications to number theory.- III. Formal power series.- 12. Formal power series and the theory of partitions.- 13. Ramanujan's congruences and identities.- IV. The circle method.- 14. Analytic theory of partitions.- 15. Application of the circle method to modular forms of positive dimension.- Editor's notes.