(late) Tsuneo ArakawaTomoyoshi IbukiyamaProfessorDepartment of MathematicsGraduate School of ScienceOsaka UniversityMachikaneyama 1-1 Toyonaka, Osaka, 560-0043 JapanMasanobu KanekoProfessorFaculty of MathematicsKyushu UniversityMotooka 744, Nishi-ku, Fukuoka, 819-0395, Japan

"The book touches on all of the well-known classical results related to Bernoulli numbers and zeta functions ... . The book will offer something to readers at all levels of expertise, from the student of number theory looking for interesting topics to delve into, to researchers looking for an overview of various results, in each case pointing the way to further study." (Luis Manuel Navas Vicente, Mathematical Reviews, October, 2015) "This book ... is perhaps the first full-length treatment of these fascinating numbers-certainly the first modern one. ... the book has an interdisciplinary character, offering thorough treatments of the Bernoulli numbers from the optics of the history of mathematics, combinatorics, analytic number theory, and algebraic number theory ... . Summing Up: Highly recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 52 (10), June, 2015) "The present book contains some specific material reflecting the research interests of the authors. ... The monograph is a useful addition to the library of every researcher working on special numbers and special functions." (Khristo N. Boyadzhiev, zbMATH 1312.11015, 2015) "The book under review is about Bernoulli numbers and zeta functions. ... The main audience for the book are researchers and students studying Bernoulli numbers and related topics. The text of the book is very fluent. Concepts and proofs are introduced in detail, and it is easy to follow for reader. There are some exercises, so the book can be used in a graduate course as well." (Mehdi Hassani, MAA Reviews, December, 2014)

• ​1. Bernoulli Numbers 2. Stirling Numbers and Bernoulli Numbers3. Theorem of Clausen and von Staudt, and Kummer’s Congruence4. Generalized Bernoulli Numbers5. Summation Formula of Euler–Maclaurin and Riemann Zeta Function 6. Quadratic Forms and Ideal Theory of Quadratic Fields 7. Congruence Between Bernoulli Numbers and Class Numbers of Imaginary Quadratic Fields 8. Character Sums and Bernoulli Numbers 9. Special Values and Complex Integral Representation of L-functions 10. Class Number Formula and an Easy Zeta Function of a Prehomogeneous Vector Space11. p-adic Measure and Kummer’s Congruence12. Hurwitz Numbers 13. The Barnes Multiple Zeta Function14. Poly-Bernoulli NumbersReferencesIndex

(late) Tsuneo ArakawaTomoyoshi IbukiyamaProfessorDepartment of MathematicsGraduate School of ScienceOsaka UniversityMachikaneyama 1-1 Toyonaka, Osaka, 560-0043 JapanMasanobu KanekoProfessorFaculty of MathematicsKyushu UniversityMotooka 744, Nishi-ku, Fukuoka, 819-0395, Japan

"The book touches on all of the well-known classical results related to Bernoulli numbers and zeta functions ... . The book will offer something to readers at all levels of expertise, from the student of number theory looking for interesting topics to delve into, to researchers looking for an overview of various results, in each case pointing the way to further study." (Luis Manuel Navas Vicente, Mathematical Reviews, October, 2015) "This book ... is perhaps the first full-length treatment of these fascinating numbers-certainly the first modern one. ... the book has an interdisciplinary character, offering thorough treatments of the Bernoulli numbers from the optics of the history of mathematics, combinatorics, analytic number theory, and algebraic number theory ... . Summing Up: Highly recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 52 (10), June, 2015) "The present book contains some specific material reflecting the research interests of the authors. ... The monograph is a useful addition to the library of every researcher working on special numbers and special functions." (Khristo N. Boyadzhiev, zbMATH 1312.11015, 2015) "The book under review is about Bernoulli numbers and zeta functions. ... The main audience for the book are researchers and students studying Bernoulli numbers and related topics. The text of the book is very fluent. Concepts and proofs are introduced in detail, and it is easy to follow for reader. There are some exercises, so the book can be used in a graduate course as well." (Mehdi Hassani, MAA Reviews, December, 2014)

• ​1. Bernoulli Numbers 2. Stirling Numbers and Bernoulli Numbers3. Theorem of Clausen and von Staudt, and Kummer’s Congruence4. Generalized Bernoulli Numbers5. Summation Formula of Euler–Maclaurin and Riemann Zeta Function 6. Quadratic Forms and Ideal Theory of Quadratic Fields 7. Congruence Between Bernoulli Numbers and Class Numbers of Imaginary Quadratic Fields 8. Character Sums and Bernoulli Numbers 9. Special Values and Complex Integral Representation of L-functions 10. Class Number Formula and an Easy Zeta Function of a Prehomogeneous Vector Space11. p-adic Measure and Kummer’s Congruence12. Hurwitz Numbers 13. The Barnes Multiple Zeta Function14. Poly-Bernoulli NumbersReferencesIndex