Download or read book Modular Functions and Dirichlet Series in Number Theory written by Tom M. Apostol and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke’s theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr’s theory of equivalence of general Dirichlet series.

Download or read book Modular Forms written by Robert Alexander Rankin and published by . This book was released on 1984 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Elliptic Modular Functions written by B. Schoeneberg and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a fully detailed introduction to the theory of modular functions of a single variable. I hope that it will fill gaps which in view ofthe lively development ofthis theory have often been an obstacle to the students' progress. The study of the book requires an elementary knowledge of algebra, number theory and topology and a deeper knowledge of the theory of functions. An extensive discussion of the modular group SL(2, Z) is followed by the introduction to the theory of automorphic functions and auto morphic forms of integral dimensions belonging to SL(2,Z). The theory is developed first via the Riemann mapping theorem and then again with the help of Eisenstein series. An investigation of the subgroups of SL(2, Z) and the introduction of automorphic functions and forms belonging to these groups folIows. Special attention is given to the subgroups of finite index in SL (2, Z) and, among these, to the so-called congruence groups. The decisive role in this setting is assumed by the Riemann-Roch theorem. Since its proof may be found in the literature, only the pertinent basic concepts are outlined. For the extension of the theory, special fields of modular functions in particular the transformation fields of order n-are studied. Eisen stein series of higher level are introduced which, in case of the dimension - 2, allow the construction of integrals of the 3 rd kind. The properties of these integrals are discussed at length.

Download or read book Modular Functions in Analytic Number Theory written by Marvin Isadore Knopp and published by American Mathematical Soc.. This book was released on 2008 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions, $\eta(\tau)$ and $\vartheta(\tau)$, and their applications to two number-theoretic functions, $p(n)$ and $r_s(n)$. They are well chosen, as at the heart of these particular applications to the treatment of these specific number-theoretic functions lies the general theory of automorphic functions, a theory of far-reaching significance with important connections to a great many fields of mathematics. The book is essentially self-contained, assuming only a good first-year course in analysis. The excellent exposition presents the beautiful interplay between modular forms and number theory, making the book an excellent introduction to analytic number theory for a beginning graduate student. Table of Contents: The Modular Group and Certain Subgroups: 1. The modular group; 2. A fundamental region for $\Gamma(1)$; 3. Some subgroups of $\Gamma(1)$; 4. Fundamental regions of subgroups. Modular Functions and Forms: 1. Multiplier systems; 2. Parabolic points; 3 Fourier expansions; 4. Definitions of modular function and modular form; 5. Several important theorems.The Modular Forms $\eta(\tau)$ and $\vartheta(\tau)$: 1. The function $\eta(\tau)$; 2. Several famous identities; 3. Transformation formulas for $\eta(\tau)$; 4. The function $\vartheta(\tau)$. The Multiplier Systems $\upsilon_{\eta}$ and $\upsilon_{\vartheta}$: 1. Preliminaries; 2. Proof of theorem 2; 3. Proof of theorem 3. Sums of Squares: 1. Statement of results; 2. Lipschitz summation formula; 3. The function $\psi_s(\tau)$; 4. The expansion of $\psi_s(\tau)$ at $-1$; 5. Proofs of theorems 2 and 3; 6. Related results. The Order of Magnitude of $p(n)$: 1. A simple inequality for $p(n)$; 2. The asymptotic formula for $p(n)$; 3. Proof of theorem 2. The Ramanujan Congruences for $p(n)$: 1. Statement of the congruences; 2. The functions $\Phi_{p, r}(\tau)$ and $h_p(\tau)$; 3. The function $s_{p, r}(\tau)$; 4. The congruence for $p(n)$ Modulo 11; 5. Newton's formula; 6. The modular equation for the prime 5; 7. The modular equation for the prime 7. Proof of the Ramanujan Congruences for Powers of 5 and 7: 1. Preliminaries; 2. Application of the modular equation; 3. A digression: The Ramanujan identities for powers of the prime 5; 4. Completion of the proof for powers of 5; 5.Start of the proof for powers of 7; 6. A second digression: The Ramanujan identities for powers of the prime 7; 7. Completion of the proof for powers of 7. Index. (CHEL/337.H

Download or read book Elliptic Curves Modular Forms and Their L functions written by Álvaro Lozano-Robledo and published by American Mathematical Soc.. This book was released on 2011 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

Download or read book Modular Functions of One Variable I IV written by Willem Kuyk and published by . This book was released on 1973 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Kronecker s Jugendtraum and Modular Functions written by Serge G. Vlăduț and published by CRC Press. This book was released on 1991 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the second half of the 19th century, Leopold Kronecker cherished a dream, his Jugendtraum, that he should see the formulation of a complete theory of complex multiplication. Kronecker's papers devoted to his Jugendtraum constitute the foundations of the arithmetical theory of modular functions. Vladut has studied the dream, and traces the development of elliptic function theory from its genesis to its most recent achievements. Included is a reprint of Kronecker's 1886 paper which presents many of the principal ideas of the arithmetical theory of modular functions. Translated from the Russian. Annotation copyrighted by Book News, Inc., Portland, OR

Download or read book Abelian Varieties with Complex Multiplication and Modular Functions written by Goro Shimura and published by Princeton University Press. This book was released on 2016-06-02 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.

Download or read book Vorlesungen Uber Die Theorie For Elliptischen Modulfunctionen written by Dr Robert Fricke and published by Legare Street Press. This book was released on 2023-07-18 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fricke's groundbreaking study of the theory of elliptic modular functions is a must-read for anyone interested in the foundations of modern mathematics. With clear explanations and insightful examples, Fricke offers a comprehensive overview of this complex and fascinating subject. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Download or read book Elliptic Integrals Elliptic Functions and Modular Forms in Quantum Field Theory written by Johannes Blümlein and published by Springer. This book was released on 2019-01-30 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book includes review articles in the field of elliptic integrals, elliptic functions and modular forms intending to foster the discussion between theoretical physicists working on higher loop calculations and mathematicians working in the field of modular forms and functions and analytic solutions of higher order differential and difference equations.

Download or read book Elliptic and Modular Functions from Gauss to Dedekind to Hecke written by Ranjan Roy and published by Cambridge University Press. This book was released on 2017-04-18 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thorough guide to elliptic functions and modular forms that demonstrates the relevance and usefulness of historical sources.

Download or read book Hypergeometric Functions My Love written by Masaaki Yoshida and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical story - of the hypergeometric functions, the configuration space of 4 points on the projective line, elliptic curves, elliptic modular functions and the theta functions - now evolves, in this book, to the story of hypergeometric funktions in 4 variables, the configuration space of 6 points in the projective plane, K3 surfaces, theta functions in 4 variables. This modern theory has been established by the author and his collaborators in the 1990's; further development to different aspects is expected. It leads the reader to a fascinating 4-dimensional world. The author tells the story casually and visually in a plain language, starting form elementary level such as equivalence relations, the exponential function, ... Undergraduate students should be able to enjoy the text.

Download or read book Korg Volca Modular The Expert Guide written by Tony Horgan and published by Tony Horgan. This book was released on 2019-10-04 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: Korg Volca Modular - The Expert Guide is a goldmine of patches, tutorials, information and inspiration. World-renowned Volca expert Tony Horgan unravels the mysteries of the Volca Modular with mind-blowing patches, clear concept visuals and straight-talking text. Across more than 200 pages, every patch point, module, button and feature is explored and explained in detail, with the help of over 60 patches for you to recreate and adapt yourself. Learn how to create musical sounds, sound effects, drones, rhythms, process other instruments with the Volca Modular reverb effect, pass another instrument through the Volca Modular, control the Volca Modular from other instruments, create random self-generating sounds, connect to audio devices, add swing effects, master the sequencer, transpose patterns, play in different scales and so much more. 218 Pages, 61 patches Contents 01 About this book * About the author, ways to make contact 02 About the Volca Modular * About the Volca Modular * Patch cables * Front panel patch point inputs and outputs * Front panel connecting lines * What is CV (Control Voltage)? 03 Patches and sounds * Talking Droid * Space Out Feedback Loop * Correct Answer * Low Frequency Sweep * Buzz Bass * Automatic Bassline * Automatic Player * Tuned Kick Bass * Brain 04 Quick-start guides * Tips and hacks * Make a "patch" * Select and play a pattern * Select and play a pattern chain * Keyboard notes * Record a pattern * Save a pattern * Record a motion sequence * Synchronise timing with other Volcas * Connect to audio devices 05 Source * About the Source module * Source Pitch Input * Trampoline Bounce - Source Pitch Input example * Source Ratio Input * 2-Tone Hooter - Source Ratio Input example * Source Fold Input * Throb - Source Fold Input example * Source Mod Input * Space Oboe - Source Mod Input example 1 * Simple Drone - Source Mod Input example 2 * Source Modulator Wave Output * Sub-bass - Source Modulator Wave Output example * Source Carrier Output * Throat Singing - Source Carrier Output example 06 Functions * About the Functions * Function 1: Attack, Hold, Release generator * Function 1 Gate Input * Self-generating Random Gate - Function 1 Gate Input example * Function 1 Attack Input * Function 1 Release Input * Function 1 Positive Output * Brass Bugle - Function 1 Positive Output example * Function 1 Inverted Output * Slow, Fast, Slow - Function 1 Inverted Output example * Function 1 End Trigger Output * Bounce Back - Function 1 End Trigger Output example * Function 2: Rise, Fall generator * Function 2: Rise, Fall generator "LFO" * Function 2 Trigger Input * Function 2 Shape Input * Release Shape-Shifter - Function 2 Shape Input example * Function 2 Time Input * Alien Visit - Function 2 Time Input example * Function 2 Positive Output * Metallic Bounce - Function 2 Positive Output example * Function 2 Inverted Output * Function 2 End Trigger Output * Computer Processing - Function 2 End Trigger Output example 07 Clock / Sequences * Clock and Sequences * Slow, Faster, Slower - Clock Offset Input example * Sequences 1/4, 1/3, 1/2 Gate Output * Accents and Polyrhythms - Sequences 1/4, 1/3, 1/2 Gate Output example * Sequences Gate Output * Sequences Gate Output example * Sequences Pitch Output * Reverb On High Notes - Sequences Pitch Output example * Portamento - Sequences Pitch Output example 2 08 Dual LPG * Dual LPG explained * LPG 1 Signal Input * Easy sub-bass - LPG 1 Signal Input example * LPG 1 Level Input * Amplitude Modulation - LPG 1 Level Input example * LPG 1 Signal Output * Pitch Explosion - LPG 1 Signal Output example * Play It Louder - LPG 1 Signal Output example 2 * LPG 2 Signal Input * Organ Bass - LPG 2 Signal Input example * LPG 2 Level Input * Time Machine - LPG 2 Level Input example * LPG 2 Signal Output * Fuzz Bass - LPG 2 Signal Output example * Cutoff 09 Split * About the Split module * Split Input 1 / Output 1-A, Output 1-B * Split Input 2 / Output 2-A, Output 2-B * Power Station - Split example * Split mixer * Lift Off - Split mixer example * Lift Off (Extra Bass) - Split mixer example 10 Utility a, b, c * Utility a, b, c * Signal Input a * Signal Input b * Signal Input c * c knob * Intensity Control - c knob example * Volume Fluctuation - Utility a, b, c example * a+bxc Output * Self-generating Sound - a+bxc Output example 1 * Basic Wave Mixer - a+bxc Output example 2 * a-bxc Output * Pulse Width Modulation - a-bxc Output example 1 * If I Was A Pitch Man - a-bxc Output example 2 11 Woggle * About the Woggle * Woggle Sample Input * Video Game Jump - Woggle Sample Input example * Noisy Reverb - Woggle Sample Output example * Woggle Trigger Input * Simple 4th Note Trigger - Woggle Trigger Input * Woggle Stepped Output * Random Release - Woggle Stepped Output example * Woggle Smooth Output * Space Bee - Woggle Smooth Output example 12 Space Out, Microtune * About the Space Out module * Space Out Signal Input * Add reverb to an external sound source * Carrier Wave Always Audible - Space Out Signal Input example * Space Out Amount knob * Space Out Amount Input * Dry / Wet Foghorn - Space Out Amount Input example * Space Out Signal Output * Rocket Explode - Space Out Signal Output example * Microtune 13 CV In, Gate In, Sync * CV Input, Gate Input * Using CV and Gate to play notes * Calibrating (tuning) the CV Input * CV Output 1 * CV Output 2 * CV Input mixer example * Sync In / Out * Sync connections * Receive swing from another Volca * Sync problems 14 Volume, audio out, headphones * Volume * Headphones Output (audio output) 15 Buttons * Play, Active Step * Record (live recording) * Step Record * Memory, Write * Func 16 Keyboard * About the keyboard * Scale * Tonic * Microtuning * Microtuning keys * Octave keys * Randomize keys * Gate * Sequencer mode: Bounce * Sequence mode: Stoch. * Motion sequence keys * Clear keys 17 Power * Power options 18 Global parameters * Global parameters * System update (firmware update) 19 Module circuit diagrams * Source circuit diagram * Clock / Sequences circuit diagram * Functions circuit diagram * Split circuit diagram * LGP circuit diagram * Utility a, b, c circuit diagram * Woggle circuit diagram * Space Out circuit diagram

Download or read book Harmonic Maass Forms and Mock Modular Forms Theory and Applications written by Kathrin Bringmann and published by American Mathematical Soc.. This book was released on 2017-12-15 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modular forms and Jacobi forms play a central role in many areas of mathematics. Over the last 10–15 years, this theory has been extended to certain non-holomorphic functions, the so-called “harmonic Maass forms”. The first glimpses of this theory appeared in Ramanujan's enigmatic last letter to G. H. Hardy written from his deathbed. Ramanujan discovered functions he called “mock theta functions” which over eighty years later were recognized as pieces of harmonic Maass forms. This book contains the essential features of the theory of harmonic Maass forms and mock modular forms, together with a wide variety of applications to algebraic number theory, combinatorics, elliptic curves, mathematical physics, quantum modular forms, and representation theory.

Download or read book Modular Forms written by Henri Cohen and published by American Mathematical Soc.. This book was released on 2017-08-02 with total page 714 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and “fun” subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin–Lehner–Li theory of newforms and including the theory of Eisenstein series, Rankin–Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms. Some “gems” of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms. This book is essentially self-contained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.

Download or read book Advances in Cryptology CRYPTO 93 written by Douglas R. Stinson and published by Springer. This book was released on 2003-05-15 with total page 502 pages. Available in PDF, EPUB and Kindle. Book excerpt: The CRYPTO ’93 conference was sponsored by the International Association for Cryptologic Research (IACR) and Bell-Northern Research (a subsidiary of Northern Telecom), in co-operation with the IEEE Computer Society Technical Committee. It took place at the University of California, Santa Barbara, from August 22-26, 1993. This was the thirteenth annual CRYPTO conference, all of which have been held at UCSB. The conference was very enjoyable and ran very of the General Chair, Paul Van Oorschot. smoothly, largely due to the efforts It was a pleasure working with Paul throughout the months leading up to the conference. There were 136 submitted papers which were considered by the Program Committee. Of these, 38 were selected for presentation at the conference. There was also one invited talk at the conference, presented by Miles Smid, the title of which was “A Status Report On the Federal Government Key Escrow System.” The conference also included the customary Rump Session, which was presided over by Whit Diffie in his usual inimitable fashion. Thanks again to Whit for organizing and running the Rump session. This year, the Rump Session included an interesting and lively panel discussion on issues pertaining to key escrowing. Those taking part were W. Diffie, J. Gilmore, S. Goldwasser, M. Hellman, A. Herzberg, S. Micali, R. Rueppel, G. Simmons and D. Weitzner.

Download or read book Modular Functions of One Variable V written by J. P. Serre and published by Springer. This book was released on 2006-11-15 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: The proceedings of the conference are being published in two parts, and the present volume is mostly algebraic (congruence properties of modular forms, modular curves and their rational points, etc.), whereas the second volume will be more analytic and also include some papers on modular forms in several variables.