Lecture notes for Math 259: Introduction to Analytic Number Theory (Spring 200[2-]3)

If you find a mistake, omission, etc., please let me know by e-mail.

The orange ball marks our current location in the course.

For an explanation of the background pattern, skip ahead to the end of the page.

plan.pdf and intro.pdf: administrivia and philosophy/examples

elem.pdf: Elementary methods I: Variations on Euclid

euler.pdf: Elementary methods II: The Euler product for s>1 and consequences (corrected 7.ii.2003)

dirichlet.pdf: Dirichlet characters and L-series; Dirichlet's theorem under the hypothesis that L-series do not vanish at s=1 (corrected 7.ii.2003; edited 11.ii.2003)

chebi.pdf: Cebysev's method; introduction of Stirling's approximation, and of the von Mangoldt function \Lambda(n) and its sum \psi(x) (corrected 13.ii.2003, and again 23.ii.2003 (see Exercise 7))

click here For Erdos' simplification of Cebysev's proof of the "Bertrand Postulate": there exists a prime between x and 2x for all x>1. Adapted from Hardy and Wright, pages 343-344.

psi.pdf: Complex analysis enters the picture via the contour integral formula for \psi(x) and similar sums (corrected 19.ii.2003)

zeta1.pdf: The functional equation for the Riemann zeta function using Poisson inversion on theta series; basic facts about \Gamma(s) as a function of a complex variable s (corrected 19.ii.2003; again 23.ii.2003 (typo in Exercise 2))

gamma.pdf: More about \Gamma(s) as a function of a complex variable s: product formula, Stirling approximation, and some consequences (corrected 19.ii.2003; edited 20.ii.2003) Exercises corrected 6.iii.2003)

Apropos Exercise 5: here's a graph of
S(x) = x - x2 + x4 - x8 + x16 - x32 + - ...
for x in [0,0.9995]. (Apply the ``magnifying glass'' to the top right corner to see the first few oscillations.) The fact that S(0.995) = 0.50088... > 1/2, together with the functional equation
S(x) = x - S(x2)
(from which S(x) = x - x2 + S(x4) > S(x4)), suffice to refute the guess that S(x) approaches 1/2 as x approaches 1.

prod.pdf: Functions of finite order: Hadamard's product formula and its logarithmic derivative (corrected 24.ii.2003)

zeta2.pdf: The Hadamard products for \xi(s) and \zeta(s); vertical distribution of the zeros of \zeta(s).

This picture appears without explanation on the web page for John Derbyshire's Prime Obsession. It is a plot of the Riemann zeta function on the boundary of the rectangle [0.4,0.6]+[0,14.5]i in the complex plane. Since the contour winds around the origin once (and does not contain the point s=1, which is the unique pole of zeta(s)), the zeta function has a unique zero inside this rectangle. Since the complex zeros are known to be symmetric about the line Re(s)=1/2, this zero must have real part exactly equal 1/2, in accordance with the Riemann hypothesis.

It is known that this first ``nontrivial zero'' of zeta(s) occurs at s=1/2+it for t=14.13472514... The pole at s=1 accounts for the wide swath in the third quadrant, which corresponds to s of imaginary part less than 1.

Here's a similar picture for L(s,\chi4) on [0.4,0.6]+[0,11]i. Without a pole in the neighborhood, this picture is less interesting visually. We see the first two nontrivial zeros, with imaginary parts 6.0209489... and 10.2437703...

For more pictures along these lines, see Juan Arias de Reyna's manuscript ``X-Ray of Riemann's Zeta function'', Part 1 and Part 2.

free.pdf: The nonvanishing of \zeta(s) on the edge \sigma=1 of the critical strip, and the classical zero-free region 1-\sigma << 1/log|t| for \zeta(s)

pnt.pdf: Conclusion of the proof of the Prime Number Theorem with error bound; the Riemann Hypothesis, and some of its consequences and equivalent statements. (corrected 6.iii.2003)

Here's a new expository paper by B. Conrey on the Riemann Hypothesis, which includes a number of further suggestive pictures involving the Riemann zeta function, its zeros, and the distribution of primes.

Here's the Rubinstein-Sarnak paper "Chebyshev's Bias" (in PostScript, from the journal Experimental Mathematics where the paper appeared in 1994).

Here's a bibliography of fast computations of \pi(x).

lsx.pdf: L(s,\chi) as an entire function [\chi a nontrivial primitive character mod q]; Gauss sums, and the functional equation relating L(s,\chi) with L(1-s,\bar\chi) [corrected March 9 and 31]

pnt_q.pdf: Product formula for L(s,\chi), and ensuing partial-fraction decomposition of its logarithmic derivative; a (bad!) zero-free region for L(s,\chi), and resulting estimates on \psi(x,\chi) and thus on \psi(x, a mod q) and \pi(x, a mod q). The Extended Riemann Hypothesis and consequences. (corrected 31.iii.2003)

free_q.pdf: The classical region 1-\sigma << 1/log(q|t|+2) free of zeros of L(s,\chi) with at most one exception \beta; the resulting asymptotics for \psi(x, a mod q) etc.; lower bounds on 1-\beta and L(1,\chi), culminating with Siegel's theorem. (corrected 31.iii.2003)

l1x.pdf: Closed formulas for L(1,chi) and their relationship with cyclotomic units, class numbers, and the distribution of quadratic residues. (corrected 31.iii.2003)

sieve.pdf: The Selberg (a.k.a. quadratic) sieve and some applications (corrected 2.iv.2003)

weyl.pdf: Introduction to exponential sums; Weyl's equidistribution theorem

kmv.pdf: Kuzmin's inequality on sum(e(c_n)) with c_n in a nearly arithmetic progression; estimates on the mean square of an exponential sum, culminating with the Montgomery-Vaughan inequality (corrected and edited 9.iv.2003, again 18.iv.2003)

Apropos Beurling's function: here's MathWorld's take on B(x), including a graph on [-3,3]; this PDF version of the graph also shows the comparison with sgn(x).

vdc.pdf: The van der Corput estimates and some applications

many_pts.pdf: How many points can a curve of genus g have over the finite field of q elements? The zeta function of a curve over a finite field; the Weil and Drinfeld-Vladut bounds, and related matters.

Here are some tables of curves of given genus over finite fields with many rational points, maintained by Gerard van der Geer and Marcel van der Vlugt.

disc.pdf: How small can |disc(K)| be for a number field K of degree n=r1+2r2?

Here is Odlyzko's list of publications, including several versions and updates of his "Bounds for discriminants and related estimates for class numbers, regulators, and zeros of zeta functions: A survey of recent results" (originally in Sém. Th. des Nombres, Bordeaux #2 (1990), 119-141).

disc.pdf: Stark's analytic lower bound on the absolute value of the discriminant of a number field (assuming GRH).

Here are some tables of number fields, compiled by Henri Cohen.
kloos.pdf: An application of Weil's bound on Kloosterman sums, which we'll use again to estimate coefficients of modular forms.

modular.pdf: Some more about modular forms for the full modular group PSL2(Z) [continuing in a few directions where Serre's A Course in Arithmetic, Chapter VII, left off].

For elementary proofs of the identities between Eisenstein series, see
Skoruppa, N.P.: A quick combinatorial proof of Eisenstein series idenities, J. Number Theory 43 (1993), 68-73
and references contained therein. [Robin Chapman (rjc@maths.exeter.ac.uk) posted this reference to sci.math.research.]
poincare.pdf: Proof of the bound O(nk-1/4+epsilon) on the coefficients of a PSL2(Z) cusp form of weight 2k, using Poincaré series and bounds on Kloosterman sums



So what's with the whorls in the background pattern? They're a visual illustration of an exponential sum, that is, sum(exp i f(n), n=1...N). Even simple functions f can give rise to interesting behavior and/or important open problems as we vary N. What function f produced the background for this page? See here for more information.