analytic number theory

There is no required textbook for the course. Instead I’ll place the following books on reserve at the library:

1. G.J.O. Jameson, The prime number theorem. Cambridge 2003 LMS Student texts vol 53.

2. T. Apostol, Introduction to analytic number theory, Undergraduate texts in Mathematics, Springer Verlag, 1976.

3. H.L. Montgomery and R.C. Vaughan, Multiplicative number theory, I. Classical theory. Cambridge university press, Cambridge studies in advanced math vol 97, 2007.

4. Mendes-France and Tenenbaum, The prime numbers and their distribution, American Math. Soc. Student Math. Library 6, 2000.

5. J. Stopple, A primer of analytic number theory, Cambridge 2003.

6. R. Ayoub, An introduction to the Analytic theory of numbers. AMS 1963.

7. H. Davenport, Multiplicative number theory. Springer GTM.

Also I’ll put up notes on this website. My aim in this course will be to discuss several problems related to the distribution of prime numbers. One high point for the course will be the proof of the prime number theorem which gives an asymptotic for the number of primes up to x. I will also discuss Riemann’s seminal paper introducing the zeta function as a tool in prime number theory, explain some of the properties of zeta, and the connection between primes and the Riemann hypothesis. Other topics may include sieves (e.g. showing upper bounds for the number of twin primes), the primality test, gaps between primes etc. I will assume that you have some knowledge of number theory already, at the level of 152, and that you’re also comfortable with analysis and thinking about the size of things, and have some familiarity with complex analysis (say, up to Cauchy’s theorem). If you’re concerned with the background, please feel free to talk to me. To brush up on complex analysis you could look at the book by Green and Krantz (Function theory of one complex variable, first four chapters), or Ahlfors (Complex Analysis, first four chapters), or Copson (An introduction to the theory of functions of a complex variable, first six chapters).

Here are some notes based on lectures from Math 152 in previous years: Bertrand’s postulate, Dirichlet’s Theorem I, Dirichlet’s Theorem II, Dirichlet’s Theorem III, Dirichlet’s Theorem IV. I won’t be assuming that you know everything here, but familiarity (or comfort) with some of the techniques mentioned here will be useful, and I’ll review some of this in the first couple of lectures.

Problem 1.

In this problem put $\sigma=1+1 / \log x$. Show that
\sum_{p>x} \log \left(1-\frac{1}{p^{\sigma}}\right)^{-1}=\sum_{p>x} \frac{1}{p^{\sigma}}+O\left(\frac{1}{x}\right)=\int_{1}^{\infty} \frac{e^{-t}}{t} d t+O\left(\frac{1}{\log x}\right)

use that $\sum_{p \leq z}(\log p) / p=\log z+O(1)$

Problem 2.

Consider the series
F(s)=\sum_{n=1}^{\infty} \frac{1}{\phi(n)^{s}}
Where does this converge absolutely? Show that $F(s)$ extends meromorphically to the region $\operatorname{Re}(s)>0$, and is analytic except a simple pole at $s=1 .$ Make a guess about how many integers $n$ are there with $\phi(n) \leq x .$ What issues would you face if you want to make a rigorous proof of your guess?

Problem 3.

For a nice smooth function $f$ with rapid decay show that
\sum_{n \in \mathbb{Z}} f(n+\alpha)=\sum_{k \in \mathbb{Z}} \hat{f}(k) e^{2 \pi i k \alpha} .
Prove that for any $x>0$
x^{-\frac{1}{2}} \sum_{n \in \mathbb{Z}} e^{-\pi(n+\alpha)^{2} / x}=\sum_{k \in \mathbb{Z}} e^{-\pi k^{2} x+2 \pi i k \alpha},
and that
-2 \pi x^{-\frac{3}{2}} \sum_{n \in \mathbb{Z}}(n+\alpha) e^{-\pi(n+\alpha)^{2} / x}=2 \pi i \sum_{k \in \mathbb{Z}} k e^{-\pi k^{2} x+2 \pi i k \alpha} .

Problem 4. Prove the functional equation
\left(\frac{5}{\pi}\right)^{s / 2} \Gamma(s / 2) L\left(s, \chi_{5}\right)=\left(\frac{5}{\pi}\right)^{(1-s) / 2} \Gamma((1-s) / 2) L\left(1-s, \chi_{5}\right),
and show that the above function is holomorphic for all $s \in \mathbb{C}$.

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