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Title: **Modular Functions in Analytic Number Theory**

Publisher: **Markham Publishing Co.**

Publication Date: **1970**

Binding: **hardcover**

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Chicago, Markham Publishing Company 1970.
(1970)

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**Book Description **Chicago, Markham Publishing Company 1970., 1970. 8°. X, 150 S. OLn. (Markham Mathematics Series). Gutes Exemplar. OLn. (Markham Mathematics Series). Bookseller Inventory # 86248CB

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Published by
Markham Publishing Company
(1970)

ISBN 10: 0841010005
ISBN 13: 9780841010000

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**Book Description **Markham Publishing Company, 1970. Book Condition: Very Good. 150 pp., Hardcover, corners rubbed, else very good. Bookseller Inventory # ZB1017828

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**Book Description ** Bookseller Inventory # 70145

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ISBN 10: 0821844881
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**Book Description **Book Condition: New. Bookseller Inventory # 20022602-n

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Amer Mathematical Society
(2008)

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**Book Description **Amer Mathematical Society, 2008. HRD. Book Condition: New. New Book. Shipped from US within 10 to 14 business days. Established seller since 2000. Bookseller Inventory # KB-9780821844885

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American Mathematical Society, United States
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**Book Description **American Mathematical Society, United States, 2008. Hardback. Book Condition: New. 2nd Revised edition. Language: English . Brand New Book. 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). Bookseller Inventory # AAS9780821844885

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Published by
American Mathematical Society
(2002)

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**Book Description **American Mathematical Society, 2002. HRD. Book Condition: New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. Bookseller Inventory # CE-9780821844885

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American Mathematical Society, United States
(2008)

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**Book Description **American Mathematical Society, United States, 2008. Hardback. Book Condition: New. 2nd Revised edition. Language: English . Brand New Book. 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). Bookseller Inventory # AAS9780821844885

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American Mathematical Society

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**Book Description **American Mathematical Society. Hardback. Book Condition: new. BRAND NEW, Modular Functions in Analytic Number Theory (2nd Revised edition), Marvin I. Knopp, 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). Bookseller Inventory # B9780821844885

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Published by
New York - 1993
(1993)

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**Book Description **New York - 1993, 1993. Chelsea publishing-company, New York - 1993, 15,50x23,50 cm, relié, XII + 156 pages Bon état. Bookseller Inventory # 23189-TF

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