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This edited volume presents a collection of carefully refereed
articles covering the latest advances in Automorphic Forms and
Number Theory, that were primarily developed from presentations
given at the 2012 "International Conference on Automorphic Forms
and Number Theory," held in Muscat, Sultanate of Oman. The present
volume includes original research as well as some surveys and
outlines of research altogether providing a contemporary snapshot
on the latest activities in the field and covering the topics of:
Borcherds products Congruences and Codes Jacobi forms Siegel and
Hermitian modular forms Special values of L-series Recently, the
Sultanate of Oman became a member of the International Mathematical
Society. In view of this development, the conference provided the
platform for scientific exchange and collaboration between
scientists of different countries from all over the world. In
particular, an opportunity was established for a close exchange
between scientists and students of Germany, Oman, and Japan. The
conference was hosted by the Sultan Qaboos University and the
German University of Technology in Oman.
Two major subjects are treated in this book. The main one is the
theory of Bernoulli numbers and the other is the theory of zeta
functions. Historically, Bernoulli numbers were introduced to give
formulas for the sums of powers of consecutive integers. The real
reason that they are indispensable for number theory, however, lies
in the fact that special values of the Riemann zeta function can be
written by using Bernoulli numbers. This leads to more advanced
topics, a number of which are treated in this book: Historical
remarks on Bernoulli numbers and the formula for the sum of powers
of consecutive integers; a formula for Bernoulli numbers by
Stirling numbers; the Clausen-von Staudt theorem on the
denominators of Bernoulli numbers; Kummer's congruence between
Bernoulli numbers and a related theory of p-adic measures; the
Euler-Maclaurin summation formula; the functional equation of the
Riemann zeta function and the Dirichlet L functions, and their
special values at suitable integers; 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; class number formula
for positive definite binary quadratic forms; congruences between
some class numbers and Bernoulli numbers; simple zeta functions of
prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta
functions and their special values; the functional equation of the
doub le zeta functions; and poly-Bernoulli numbers. An appendix by
Don Zagier on curious and exotic identities for Bernoulli numbers
is also supplied. This book will be enjoyable both for amateurs and
for professional researchers. Because the logical relations between
the chapters are loosely connected, readers can start with any
chapter depending on their interests. The expositions of the topics
are not always typical, and some parts are completely new.
This edited volume presents a collection of carefully refereed
articles covering the latest advances in Automorphic Forms and
Number Theory, that were primarily developed from presentations
given at the 2012 "International Conference on Automorphic Forms
and Number Theory," held in Muscat, Sultanate of Oman. The present
volume includes original research as well as some surveys and
outlines of research altogether providing a contemporary snapshot
on the latest activities in the field and covering the topics of:
Borcherds products Congruences and Codes Jacobi forms Siegel and
Hermitian modular forms Special values of L-series Recently, the
Sultanate of Oman became a member of the International Mathematical
Society. In view of this development, the conference provided the
platform for scientific exchange and collaboration between
scientists of different countries from all over the world. In
particular, an opportunity was established for a close exchange
between scientists and students of Germany, Oman, and Japan. The
conference was hosted by the Sultan Qaboos University and the
German University of Technology in Oman.
Two major subjects are treated in this book. The main one is the
theory of Bernoulli numbers and the other is the theory of zeta
functions. Historically, Bernoulli numbers were introduced to give
formulas for the sums of powers of consecutive integers. The real
reason that they are indispensable for number theory, however, lies
in the fact that special values of the Riemann zeta function can be
written by using Bernoulli numbers. This leads to more advanced
topics, a number of which are treated in this book: Historical
remarks on Bernoulli numbers and the formula for the sum of powers
of consecutive integers; a formula for Bernoulli numbers by
Stirling numbers; the Clausen von Staudt theorem on the
denominators of Bernoulli numbers; Kummer's congruence between
Bernoulli numbers and a related theory of "p"-adic measures; the
Euler Maclaurin summation formula; the functional equation of the
Riemann zeta function and the Dirichlet L functions, and their
special values at suitable integers; 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; class number formula
for positive definite binary quadratic forms; congruences between
some class numbers and Bernoulli numbers; simple zeta functions of
prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta
functions and their special values; the functional equation of the
double zeta functions; and poly-Bernoulli numbers. An appendix by
Don Zagier on curious and exotic identities for Bernoulli numbers
is also supplied. This book will be enjoyable both for amateurs and
for professional researchers. Because the logical relations between
the chapters are loosely connected, readers can start with any
chapter depending on their interests. The expositions of the topics
are not always typical, and some parts are completely new."
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