Topics in Analytic Number Theory

1. Front Matter

   Pages N1-IX

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2. Analytic Tools

   1. Bernoulli Polynomials and Bernoulli Numbers

      • Hans Rademacher

      Pages 1-13

   2. The Euler-MacLaurin Sum Formula

      • Hans Rademacher

      Pages 14-30

   3. The Γ-function and Mellin’s Theorem

      • Hans Rademacher

      Pages 30-58

   4. The Phragmén-Lindelöf Theorem

      • Hans Rademacher

      Pages 58-70

   5. The Poisson Sum Formula and Applications

      • Hans Rademacher

      Pages 70-79

3. Special Functions

   1. The Riemann ζ-function

      • Hans Rademacher

      Pages 80-92

   2. About the Prime-number Theorem and the Zeros of the ζ-function

      • Hans Rademacher

      Pages 93-116

   3. The Eisenstein Series

      • Hans Rademacher

      Pages 117-137

   4. The Transformation of log η(τ) and the Theory of the Dedekind Sums

      • Hans Rademacher

      Pages 137-163

   5. The ϑ-functions

      • Hans Rademacher

      Pages 164-183

   6. Elliptic Functions and their Applications to Number Theory

      • Hans Rademacher

      Pages 183-208

4. Formal Power Series

   1. Formal Power Series and the Theory of Partitions

      • Hans Rademacher

      Pages 209-237

   2. Ramanujan’s Congruences and Identities

      • Hans Rademacher

      Pages 237-263

5. The Circle Method

   1. Analytic Theory of Partitions

      • Hans Rademacher

      Pages 264-302

   2. Application of the Circle Method to Modular Forms of Positive Dimension

      • Hans Rademacher

      Pages 303-313

6. Back Matter

   Pages 314-322

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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 ........... .

• analytic number theory
• binomial
• number theory

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