Titlebook: Bernoulli Numbers and Zeta Functions; Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko Book 2014 Springer Japan 2014 Bernoulli numbers a

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Bernoulli Numbers,e Bernoulli numbers in connection to the study of the sums of powers of consecutive integers .. After listing the formulas for the sums of powers. up to . = 10 (Bernoulli expresses the right-hand side without factoring), he gives a general formula involving the numbers which are known today as Berno

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,Theorem of Clausen and von Staudt, and Kummer’s Congruence,l part” of .. is given by the following theorem. This result gives a foundation for studying .-adic properties of the Bernoulli numbers. It also plays a fundamental role in the theory of .-adic modular forms through the Eisenstein series [82].

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Generalized Bernoulli Numbers, Dirichlet character, which we define at the beginning of the first section. Bernoulli polynomials are generalizations of Bernoulli numbers with an indeterminate. These two generalizations are related, and they will appear in various places in the following chapters.

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Character Sums and Bernoulli Numbers,ormulas between exponential sums or character sums and Bernoulli numbers. We often encounter such formulas when we compare the dimension formulas of modular forms obtained by the Riemann–Roch theorem and by the trace formula. Often, the exponential sums appear in the first method and the Bernoulli n

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Class Number Formula and an Easy Zeta Function of the Space of Quadratic Forms, so-called prehomogeneous vector spaces. We also prove a class number formula of imaginary quadratic fields. Before that, we review the theory of multiplicative structure of ideals of quadratic field without proof.

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Book 2014he Riemann zeta function and the Dirichlet L functions, and their special values at suitableintegers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; c

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