- Lecturer
- Ian Nicholas Petrow
- Coordinator
- Seraina Wachter
- Classes
- Wed 8:00-10:00 (HG E 22) and Th 15:00-17:00 (HG D 1.1)

Linear algebra, modern algebra, complex analysis. Some familiarity with the Fourier series and the Fourier transform.

**New:** There will be an office hour by the assistant every week on Thursday from 13h00 to 14h00 in room HG J 15.1 (or HG J 14.1).

Date | Format | Comment |
---|---|---|

Wednesday 19.09.2018 | Lecture | Introduction; prime numbers and Euler's method |

Thursday 20.09.2018 | Lecture | Euler's method and Chebyshev's method |

Wednesday 26.09.2018 | Exercise | Exercises. Solutions. |

Thursday 27.09.2018 | Lecture | Approximation by an integral, integration by parts, intro to Dirichlet convolution |

Wednesday 3.10.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | The Möbius inversion formula, application to prime numbers. Exercises. Solutions. |

Thursday 4.10.2018 | Lecture | Multiplicative functions, introduction to Dirichlet series |

Wednesday 10.10.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Abscissa of convergence and convolutions. Exercises. Solutions. |

Thursday 11.10.2018 | Lecture | Dirichlet Series and multiplicative functions |

Wednesday 17.10.2018 | Lecture | Primes in arithmetic progressions, characters of a finite abelian group |

Thursday 18.10.2018 | Lecture 15h15 to 16h00; exercise 16h15 to 17h00 | Characters of a finite abelian group. Exercises. Solutions. |

Wednesday 24.10.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Dirichlet L-functions and their analytic continuation. Exercises. Solutions. |

Thursday 25.10.2018 | Lecture | Proof of Mertens theorem in arithmetic progressions, non-vanishing of L(1,χ). |

Wednesday 31.10.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Finish L(1,χ) non-zero. Exercises. Solutions. |

Thursday 01.11.2018 | Lecture | Riemann's memoir, Fourier transform. |

Wednesday 07.11.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Fourier transform, Mellin transform. Exercises. Solutions. |

Thursday 08.11.2018 | Lecture | Mellin transform continued. Functional equation of the zeta function. |

Wednesday 14.11.2018 | Exercise | Exercises. Solutions. |

Thursday 15.11.2018 | Lecture | Primitive characters, Gauss sums, begin functional equation of L(s,χ) |

Wednesday 21.11.2018 | Exercise | Exercises. Solutions. |

Thursday 22.11.2018 | Lecture | Finish functional equation of L(s,χ), functions of bounded order. |

Wednesday 28.11.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Jensen's formula. Exercises. Solutions. |

Thursday 29.11.2018 | Lecture | Hadamard factorization theorem. |

Wednesday 05.12.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Counting zeros of the zeta function Exercises. Solutions. |

Thursday 06.12.2018 | Lecture | Weil's explicit formula |

Wednesday 12.12.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Zero counting for L(s,χ). The pnt assuming a zero-free region. Exercises. Solutions. |

Thursday 13.12.2018 | Lecture (by Prof. Nelson) | Zero free regions for the zeta function and L(s,χ) |

Wednesday 19.12.2018 | Lecture 8h15 to 9h00; exercise 9h15 to 10h00 | Warm up to Siegel's theorem. Exercises. Solutions. |

Thursday 20.12.2018 | Lecture | Siegel's theorem and the prime number theorem in arithmetic progressions |

The new exercises will be posted here on Fridays. We expect you to look at the problems and to prepare questions for the exercise class on the following week. You have the option to submit written solutions to the exercises on Thursday in the lecture to get comments on your work. These will be returned the following Thursday lecture. Solutions and a new exercise sheet will be posted the following Friday.

- - Multiplicative Number Theory, Third Edition by H. Davenport (Springer 2000)
- - Multiplicative Number Theory I. Classical Theory by H.L. Montgomery and R.C. Vaughan (Cambridge 2006)
- - Course notes "245a Analytic Prime Number Theory" by T. Tao. See T. Tao's Blog.

- - The proof of Theorem 9.3 (the quantitative zero-free region for Dirichlet L-functions) was revised.
- - Fixed a typo in the definition of the Gauss sum in section 4.2.
- - p. 17: in Ex. 2.13.3 Λ(n) changed to Λ(d).
- - End of p. 27: “has be modified” changed to “… has been modified”.
- - p. 28: "s with Re(s) > σ” changed to "s with Re(s) ≥ σ”.
- - p. 36.The index of the first sum on the page changed to “1 < n ≤ X” instead of “1 ≤ n ≤ X”
- - p. 44: e(nz) changed to e(nx)
- - p. 47: (g exp(σ x))^ (t/2π) changed to (g exp(σ x))^ (-t/2π) in two places.
- - p. 48: In the 8th line of the Proof of Thm. 6.8 the word “inequality” was changed to “equality”.
- - p. 64: In the sum after “Putting these facts together, we find …” two instances of “Re” were deleted.
- - p.69: In the proof of Thm. 9.2 ζ has a simple pole at s=1, not s-1.