18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
The functional equation for the Riemann zeta function
In this unit, we establish the functional equation prop erty for the Riemann zeta function,
which will imply its meromorphic continuation
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
von Mangoldts formula
In this unit, we derive von Mangoldts formula estimating (x) x in terms of the
critical zeroes of the Riemann zeta function. This nishes the derivation of a form of the
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
More on the zeroes of
In this unit, we derive some results about the location of the zeroes of the Riemann zeta
function, including a small zero-free region inside the critical strip.
1
Orde
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
Error bounds in the prime number theorem in arithmetic progressions
In this unit, we summarize how to derive a form of the prime number theorem in arith
metic progressions with an appropriate
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
Revisiting the sieve of Eratosthenes
This unit begins the second part of the course, in which we will investigate a class of
methods in analytic number theory known as sieves. (For non-native
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
Applying the Selberg sieve
Here are some suggestions about how to apply the Selberg sieve; this should help with
some of the exercises on the previous handout (the bound on twin primes, and t
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
Bruns combinatorial sieve
In this unit, we describe a more intricate version of the sieve of Eratosthenes, introduced
by Viggo Brun in order to study the Goldbach conjecture and the twin prim
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
Error bounds in the prime number theorem
In this unit, we introduce (without proof for now) a formula which relates the distribution
of primes to the zeroes of the Riemann zeta function. Give
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
The functional equations for Dirichlet L-functions
In this unit, we establish the functional equation prop erty for Dirichlet L-functions. Much
of the work is left as exercises.
1
Even charac
18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
The Selberg sieve
1
Review of notation
Let f : N C be an arithmetic function, and suppose we want to estimate the sum of f
over primes. More precisely, let P be a set of primes, and put
P (z