Barry Mazur is Gerhard Gade University Professor of Mathematics at Harvard University, Massachusetts. He is the author of Imagining Numbers: (Particularly the Square Root of Minus Fifteen) and co-editor, with Apostolos Doxiadis, of Circles Disturbed: The Interplay of Mathematics and Narrative. William Stein is Professor of Mathematics at the University of Washington. Author of Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach, he is also the founder of the Sage mathematical software project.

'This is an extraordinary book, really one of a kind. Written by two supreme experts, but aimed at the level of an undergraduate or curious amateur, it emphasizes the really powerful ideas, with the bare minimum of math notation and the maximum number of elegant and suggestive visuals. The authors explain why this legendary problem is so beautiful, why it is difficult, and why you should care.' Will Hearst, Hearst Corporation

• 1. Thoughts about numbers; 2. What are prime numbers?; 3. 'Named' prime numbers; 4. Sieves; 5. Questions about primes; 6. Further questions about primes; 7. How many primes are there?; 8. Prime numbers viewed from a distance; 9. Pure and applied mathematics; 10. A probabilistic 'first' guess; 11. What is a 'good approximation'?; 12. Square root error and random walks; 13. What is Riemann's hypothesis?; 14. The mystery moves to the error term; 15. Césaro smoothing; 16. A view of Li(X) - π(X); 17. The prime number theorem; 18. The staircase of primes; 19. Tinkering with the staircase of primes; 20. Computer music files and prime numbers; 21. The word 'spectrum'; 22. Spectra and trigonometric sums; 23. The spectrum and the staircase of primes; 24. To our readers of part I; 25. Slopes and graphs that have no slopes; 26. Distributions; 27. Fourier transforms: second visit; 28. Fourier transform of delta; 29. Trigonometric series; 30. A sneak preview; 31. On losing no information; 32. Going from the primes to the Riemann spectrum; 33. How many θi's are there?; 34. Further questions about the Riemann spectrum; 35. Going from the Riemann spectrum to the primes; 36. Building π(X) knowing the spectrum; 37. As Riemann envisioned it; 38. Companions to the zeta function.

Barry Mazur is Gerhard Gade University Professor of Mathematics at Harvard University, Massachusetts. He is the author of Imagining Numbers: (Particularly the Square Root of Minus Fifteen) and co-editor, with Apostolos Doxiadis, of Circles Disturbed: The Interplay of Mathematics and Narrative. William Stein is Professor of Mathematics at the University of Washington. Author of Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach, he is also the founder of the Sage mathematical software project.

'This is an extraordinary book, really one of a kind. Written by two supreme experts, but aimed at the level of an undergraduate or curious amateur, it emphasizes the really powerful ideas, with the bare minimum of math notation and the maximum number of elegant and suggestive visuals. The authors explain why this legendary problem is so beautiful, why it is difficult, and why you should care.' Will Hearst, Hearst Corporation

• 1. Thoughts about numbers; 2. What are prime numbers?; 3. 'Named' prime numbers; 4. Sieves; 5. Questions about primes; 6. Further questions about primes; 7. How many primes are there?; 8. Prime numbers viewed from a distance; 9. Pure and applied mathematics; 10. A probabilistic 'first' guess; 11. What is a 'good approximation'?; 12. Square root error and random walks; 13. What is Riemann's hypothesis?; 14. The mystery moves to the error term; 15. Césaro smoothing; 16. A view of Li(X) - π(X); 17. The prime number theorem; 18. The staircase of primes; 19. Tinkering with the staircase of primes; 20. Computer music files and prime numbers; 21. The word 'spectrum'; 22. Spectra and trigonometric sums; 23. The spectrum and the staircase of primes; 24. To our readers of part I; 25. Slopes and graphs that have no slopes; 26. Distributions; 27. Fourier transforms: second visit; 28. Fourier transform of delta; 29. Trigonometric series; 30. A sneak preview; 31. On losing no information; 32. Going from the primes to the Riemann spectrum; 33. How many θi's are there?; 34. Further questions about the Riemann spectrum; 35. Going from the Riemann spectrum to the primes; 36. Building π(X) knowing the spectrum; 37. As Riemann envisioned it; 38. Companions to the zeta function.

Barry Mazur is Gerhard Gade University Professor of Mathematics at Harvard University, Massachusetts. He is the author of Imagining Numbers: (Particularly the Square Root of Minus Fifteen) and co-editor, with Apostolos Doxiadis, of Circles Disturbed: The Interplay of Mathematics and Narrative. William Stein is Professor of Mathematics at the University of Washington. Author of Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach, he is also the founder of the Sage mathematical software project.

'This is an extraordinary book, really one of a kind. Written by two supreme experts, but aimed at the level of an undergraduate or curious amateur, it emphasizes the really powerful ideas, with the bare minimum of math notation and the maximum number of elegant and suggestive visuals. The authors explain why this legendary problem is so beautiful, why it is difficult, and why you should care.' Will Hearst, Hearst Corporation

• 1. Thoughts about numbers; 2. What are prime numbers?; 3. 'Named' prime numbers; 4. Sieves; 5. Questions about primes; 6. Further questions about primes; 7. How many primes are there?; 8. Prime numbers viewed from a distance; 9. Pure and applied mathematics; 10. A probabilistic 'first' guess; 11. What is a 'good approximation'?; 12. Square root error and random walks; 13. What is Riemann's hypothesis?; 14. The mystery moves to the error term; 15. Césaro smoothing; 16. A view of Li(X) - π(X); 17. The prime number theorem; 18. The staircase of primes; 19. Tinkering with the staircase of primes; 20. Computer music files and prime numbers; 21. The word 'spectrum'; 22. Spectra and trigonometric sums; 23. The spectrum and the staircase of primes; 24. To our readers of part I; 25. Slopes and graphs that have no slopes; 26. Distributions; 27. Fourier transforms: second visit; 28. Fourier transform of delta; 29. Trigonometric series; 30. A sneak preview; 31. On losing no information; 32. Going from the primes to the Riemann spectrum; 33. How many θi's are there?; 34. Further questions about the Riemann spectrum; 35. Going from the Riemann spectrum to the primes; 36. Building π(X) knowing the spectrum; 37. As Riemann envisioned it; 38. Companions to the zeta function.