Lectures on the Riemann zeta function / H. Iwaniec.
The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting...
Uniform Title: | University lecture series (Providence, R.I.) ;
62. |
---|---|
Main Author: | |
Language: | English |
Published: |
Providence, Rhode Island :
American Mathematical Society,
[2014]
|
Series: | University lecture series (Providence, R.I.) ;
62. |
Subjects: | |
Physical Description: | vii, 119 pages : illustrations ; 26 cm. |
Variant Title: |
Riemann zeta function |
Format: | Book |
MARC
LEADER | 00000cam a22000004i 4500 | ||
---|---|---|---|
001 | in00005631883 | ||
003 | OCoLC | ||
005 | 20220616160311.0 | ||
008 | 140602t20142014riua b 001 0 eng | ||
010 | |a 2014021164 | ||
020 | |a 9781470418519 |q (alk. paper) | ||
020 | |a 1470418517 |q (alk. paper) | ||
035 | |a (OCoLC)881064777 | ||
040 | |a DLC |b eng |e rda |c DLC |d OCLCO |d YDXCP |d BTCTA |d OCLCF |d NLGGC |d IPL |d NDD |d OCLCQ |d ORE |d HEBIS |d EEM |d UtOrBLW | ||
042 | |a pcc | ||
049 | |a EEMR | ||
050 | 0 | 0 | |a QA351 |b .I93 2014 |
082 | 0 | 0 | |a 515/.56 |2 23 |
100 | 1 | |a Iwaniec, Henryk. |0 http://id.loc.gov/authorities/names/n85289318 | |
245 | 1 | 0 | |a Lectures on the Riemann zeta function / |c H. Iwaniec. |
246 | 3 | 0 | |a Riemann zeta function |
264 | 1 | |a Providence, Rhode Island : |b American Mathematical Society, |c [2014] | |
264 | 4 | |c ©2014 | |
300 | |a vii, 119 pages : |b illustrations ; |c 26 cm. | ||
336 | |a text |b txt |2 rdacontent | ||
337 | |a unmediated |b n |2 rdamedia | ||
338 | |a volume |b nc |2 rdacarrier | ||
490 | 1 | |a University lecture series ; |v volume 62 | |
504 | |a Includes bibliographical references (page 117) and index. | ||
505 | 0 | |a Part 1. Classical Topics -- 1. Panorama of Arithmetic Functions -- 2. The Euler-Maclaurin Formula -- 3. Tchebyshev's Prime Seeds -- 4. Elementary Prime Number Theorem -- 5. The Riemann Memoir -- 6. The Analytic Continuation -- 7. The Functional Equation -- 8. The Product Formula over the Zeros -- 9. The Asymptotic Formula for N(T) -- 10. The Asymptotic Formula for ?(x) -- 11. The Zero-free Region and the PNT -- 12. Approximate Functional Equations -- 13. The Dirichlet Polynomials -- 14. Zeros off the Critical Line -- 15. Zeros on the Critical Line -- Part 2. The Critical Zeros after Levinson -- 16. Introduction -- 17. Detecting Critical Zeros -- 18. Conrey's Construction -- 19. The Argument Variations -- 20. Attaching a Mollifier -- 21. The Littlewood Lemma -- 22. The Principal Inequality -- 23. Positive Proportion of the Critical Zeros -- 24. The First Moment of Dirichlet Polynomials -- 25. The Second Moment of Dirichlet Polynomials -- 26. The Diagonal Terms -- 27. The Off-diagonal Terms -- 28. Conclusion -- 29. Computations and the Optimal Mollifier -- Appendix A. Smooth Bump Functions -- Appendix B. The Gamma Function. | |
520 | |a The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context. | ||
600 | 1 | 0 | |a Riemann, Bernhard, |d 1826-1866. |0 http://id.loc.gov/authorities/names/n81005064 |
650 | 0 | |a Functions, Zeta. |0 http://id.loc.gov/authorities/subjects/sh85052354 | |
650 | 0 | |a Riemann hypothesis. |0 http://id.loc.gov/authorities/subjects/sh2005000907 | |
650 | 0 | |a Numbers, Prime. |0 http://id.loc.gov/authorities/subjects/sh85093218 | |
600 | 1 | 7 | |a Riemann, Bernhard, |d 1826-1866. |2 fast |0 (OCoLC)fst00066508 |
650 | 7 | |a Functions, Zeta. |2 fast |0 (OCoLC)fst00936136 | |
650 | 7 | |a Numbers, Prime. |2 fast |0 (OCoLC)fst01041241 | |
650 | 7 | |a Riemann hypothesis. |2 fast |0 (OCoLC)fst01737612 | |
650 | 7 | |a Number theory |x Multiplicative number theory |x Distribution of primes. |2 msc | |
650 | 7 | |a Number theory |x Multiplicative number theory |x Asymptotic results on arithmetic functions. |2 msc | |
650 | 7 | |a Riemannsche Zetafunktion. |2 gnd | |
830 | 0 | |a University lecture series (Providence, R.I.) ; |v 62. |0 http://id.loc.gov/authorities/names/n88540797 | |
907 | |y .b12157779x |b 170816 |c 170213 | ||
998 | |a rs |b 170213 |c m |d a |e - |f eng |g riu |h 0 |i 2 | ||
994 | |a C0 |b EEM | ||
999 | f | f | |i e9558575-00bd-5667-9910-a4e461e435fa |s 0377ea34-1bd3-5daa-bd77-da2f508a36d4 |t 0 |
952 | f | f | |p Can Circulate |a Michigan State University-Library of Michigan |b Michigan State University |c MSU Remote Storage |d MSU Remote Storage |t 0 |e QA351 .I93 2014 |h Library of Congress classification |i Printed Material |m 31293035334345 |n 1 |