Prime Number Theorem

Question 0

Prove the following LEMMA:
Let χ be a character mod q induced by χ’ a character mod q’ (admit the case χ’=1). Then summing in each case over primes p=(p)•logp +O(logp)

Answer 0

Proof:
The difference is bounded by:
Σlogp where we sum over p<logq

Question 1

What is the explicit formula for the sum of the log of the primes less than x?
What about the weighted sum by the character χ?

Answer 1

Let 2d/2logx where d is the small constant needed to limit the number of zeros of such L functions.

Question 2

Prove the following LEMMA:

Let 1<d/logT and |γ|<T

Answer 2

Proof:

This sum is less than the sum when we just sum over ρ with |γ|<n+1}

Question 3

State the prime number theorem

Answer 3

PNT:
There exists c>0 such that
Σlogp = x - Ο(x•exp(-c•sqr(logx)))
Where sine is over p<x, and/

π(x)=INT(dt/logt)+
Ο(x•exp(-c•sqr(logx)))

Where integral is from 2 to x

Question 4

State the PNT for AP’s

Sketch a proof

Answer 4

PNT in AP’s:

There is a constant c>0 such thy for all q1-δ/logq. Let A>0. If q0 such that we get the same sum, without the middle term and replace c with cA in formula above.

Sketch of proof:
We use orthogonality of character to say the sum is:
=1/φ(q)•Σχ*(a)•Σχ(p)•logp
Where the first sum is over characters mod q and the second is over primes less than x.
Let χ’ be the character inducing χ and edit the term χ(p) introducing a certain error term.
Now use the explicit formula and bound the sum over zeros, setting qs theorem to bound exceptional zero term.