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.
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.
Jan 5: Overview of course. Beginning of Euler-Maclaurin.
Jan 7: Euler-Maclaurin formula sketch, see exercise 5 of problem set 1. The idea of partial summation (see Lemma 1 in Dirichlet's Theorem IV above). Applications to the harmonic sum and Stirling's formula. Big Oh, little oh, asymptotic, less than less than notations.
Jan 12: Chebyshev's estimates for psi(x) and theta(x). Equivalent forms of the prime number theorem. The asymptotic formulas for \sum_{p\le x} \log p/p and \sum_{p\le x} 1/p.
Jan 14: The mean and the variance of omega(n). The normal order of omega(n) is asymptotically log log n. The average value of the divisor function. The hyperbola method and Dirichlet's divisor problem.
Jan 19: Notes
Jan 21: Finishing the equivalence of PNT and estimates for partial sums of the Mobius function: see Notes from Jan 19. Calculating asymptotics, examples: number of square-free integers up to x, counting numbers up to x coprime to a number q.
Jan 26: More discussion on the number of integers up to x that are coprime to q. The divisor function is at most C(\epsilon)n^{\epsilon} for any positive epsilon. \phi(q)/q is bounded below by (e^{-\gamma}+o(1))/\log \log q. Mertens's theorem (did not determine that the constant is e^{-\gamma}; just stated). Four unattackable problems on primes + discussion of sieves.
Jan 28: Selberg's identity for numbers that are prime, or are composed of two prime factors. If a= \liminf \psi(x)/x and A= \limsup \psi(x)/x then a+A =2.
Feb 2: Midterm, see exam and solutions above.
Feb 4: Dirichlet series, Euler products. Definition and discussion of absolute convergence of products. Discussion of analytic functions and analytic continuation. The analytic continuation of zeta(s) to \sigma >0.
Feb 9: Discussion of the series for log zeta(s), the logarithmic derivative of zeta(s) etc. How to calculate the zeta function, say for s=1/2+it.
Feb 11: Review of contour integration and Cauchy's theorem and residue theorem. Perron's formula stated, and an overview of applications to finding asymptotics.
Feb 16: Proof of a quantitative form of Perron's formula. Asymptotic for the sum of the k-divisor function. One key step is knowing bounds for zeta(s).
Feb 18: More asymptotics. The asymptotic for the sum of d(n)^2. Comparing Dirichlet series with an appropriate power of zeta(s) to get analytic continuation. Variants of Perron's formula; e.g. integrating y^s/s^2, or y^s/(s(s+1)). Advantage: integrals are absolutely convergent; disadvantage: we get asymptotics for weighted sums and maybe some people don't like the weights.
Feb 23: Work towards the prime number theorem. Idea to get cancellation in the sum (over n up to x) of \mu(n) (\log n)^k (log x/n). Set up as a contour integral on the line c=1+1/\log x. Need to bound the generating function which is (1/zeta(s))^{(k)} (k-th derivative). Expanding out the derivative we get a numerator which is an expression involving the derivatives of zeta with order up to k, and a denominator of zeta^{k+1}. Need an upper bound for the numerator (easy), and a lower bound for the denominator (hard). Lower bound related to zeros of zeta(s). Proof of the lower bound using 3+4cos (theta) + cos(2theta) \ge 0, and the deduction that zeta(1+it) is not zero.
Feb 25: Deducing bounds for the sum of \mu(n)(\log n)^k (\log x/n). Deducing bounds for sum of \mu(n). Proof of the prime number theorem completed!
Mar 2: The functional equation for zeta(s). Definition and properties of the Gamma-function. Statement of the relation connecting theta(t) and theta(1/t) (theta(t)= \sum_{n\in Z} e^{-\pi n^2 t}). Poisson summation formula; recap of Fourier series and Fourier transforms.
Mar 4: Application of Poisson summation to prove the formula for theta. Proof of the functional equation. Properties of zeta(s); location of its trivial zeros; the symmetry of its zeros about the real axis and about the line Re(s)=1/2. How does one numerically check the Riemann hypothesis; finding sign changes of a real valued function on the line Re(s)=1/2.
Mar 9:
Mar 11: