This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.
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This modern introduction to the analytic techniques used in the investigation of zeta-function avoids technical results and the customary function-theoretic approach. Since Riemann's introduction, many other classes of "zeta-function" have been introduced and are now intensively studied.
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41 of 41 people found the following review helpful
Format:Paperback
Zeta function. This book contains a lot of application, theory, and, to my surprise, several practice problems at the end of each section to maximize the learning experience. The chapters are concise and the mathematics is relatively easy follow for those with some experience in special functions.
That, however, is the major thing to note: one needs some experience with special functions in order to find this material accessible. (Obviously, right? Otherwise one wouldn't be buying this book! However, much of this material is beyond the grasp of the average mathematics student that stopped at a bachelor's degree.)
Although this book is called an introduction, I don't think that view is entirely appropriate. The material is quite extensive, and the historocity of the zeta function and its development were kept to a minimum. The precursors to the zeta function and its development by Euler and Riemann (especially Euler's original proof to the Basel Problem) are fantastic. If you're interested in Euler's role in the development, I would look to Dunham's Euler: The Master of Us All, and for Riemann, one should turn to Edwads' Riemann's Zeta Function to read Riemann's original paper.
If you're looking for depth, conciseness, and a broad view of Riemann's zeta function, this book should suit your purposes. If you want a more historical view, I would suggest either of the other books I've mentioned, and not this one.
That, however, is the major thing to note: one needs some experience with special functions in order to find this material accessible. (Obviously, right? Otherwise one wouldn't be buying this book! However, much of this material is beyond the grasp of the average mathematics student that stopped at a bachelor's degree.)
Although this book is called an introduction, I don't think that view is entirely appropriate. The material is quite extensive, and the historocity of the zeta function and its development were kept to a minimum. The precursors to the zeta function and its development by Euler and Riemann (especially Euler's original proof to the Basel Problem) are fantastic. If you're interested in Euler's role in the development, I would look to Dunham's Euler: The Master of Us All, and for Riemann, one should turn to Edwads' Riemann's Zeta Function to read Riemann's original paper.
If you're looking for depth, conciseness, and a broad view of Riemann's zeta function, this book should suit your purposes. If you want a more historical view, I would suggest either of the other books I've mentioned, and not this one.
9 of 9 people found the following review helpful
Format:Paperback
This introductory textbook gives a very good insight of analytic number theory through the special topic of Riemann Zeta-Function. It also includes of course an excellent exposition of its relationship with the Prime Number Theorem and with Riemann and Lindelöf Hypotheses. Moreover, seven appendices provide analytic technical complements needed in the core of the text. Many well-chosen exercises and problems accompany each chapter. I use this textbook in my course on Zeta-Function with great success.
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Inside This Book
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First Sentence:
In this chapter it will be shown how the Poisson Summation Formula leads to the functional equation for the zeta-function. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
approximate functional equation, prime number theory, bounded total variation, summation formula, meromorphic function, exponential sums, explicit formulae, bounded variation Key Phrases - Capitalized Phrases (CAPs): (learn more)
Riemann Hypothesis, Lindelöf Hypothesis, Poisson Summation Formula, Cauchy's Theorem, Prime Number Theorem, Stirling's Formula, Use Exercise, Mean Value Theorem, Euler Product, Jensen's Theorem, Euler's Constant, Fubini's Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In this chapter it will be shown how the Poisson Summation Formula leads to the functional equation for the zeta-function. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
approximate functional equation, prime number theory, bounded total variation, summation formula, meromorphic function, exponential sums, explicit formulae, bounded variation Key Phrases - Capitalized Phrases (CAPs): (learn more)
Riemann Hypothesis, Lindelöf Hypothesis, Poisson Summation Formula, Cauchy's Theorem, Prime Number Theorem, Stirling's Formula, Use Exercise, Mean Value Theorem, Euler Product, Jensen's Theorem, Euler's Constant, Fubini's Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 11
books:
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17
books
cite this book:
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- Number Theory by Helmut Hasse in Back Matter
- Divergent Series (AMS Chelsea Publishing) by G. H. Hardy in Back Matter
- An Introduction to the Theory of Numbers (Oxford Science Publications) by G. H. Hardy in Back Matter
- Number Theory: An approach through history from Hammurapi to Legendre by André Weil in Back Matter
- Basic Number Theory (Classics in Mathematics) by Andre Weil in Back Matter
- Exercises for Fourier Analysis by Thomas William Korner on page 365, and Back Matter
- Exercises in Fourier Analysis by T. W. Körner on page 365, and Back Matter
- Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire on page 225, and Back Matter
- An Introduction to the Langlands Program by Joseph Bernstein on page 195
- An Introduction to Number Theory (Graduate Texts in Mathematics) by G. Everest in Back Matter
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