Linnik's Theorem Flashcards
State linnik's Theorem. Define N(α,Τ:χ)
There exists effectively computable constants c1 & c2 such that whenever (a,q)=1 there exists a prime pΞa mod q with pα,
|γ|<T}
What is the generalised Riemann hypothesis? If this were true how would this give us an asymptotic formula for the sum over all p<x with pΞa mod q of: Σlogp ?
That the completed L-functions Λ(s,χ) is zero only at points where Re(s)=1/2
Plugging this bound for the sum over zeroes in our proof of PNT in AP’s and hooding T=sqr(x) would give the asymptotic formula for the sum as:
x/φ(q) + O(sqr(x)•(logx)^2)
What is Bombieri’s theorem?
What is a theorem of this kind called?
There is a constant c>0 such that for all T and 1/2=, since they show there are relatively few zeros in some region to the right of the half (we hope there should be none)
Use Bombieri’s theorem to show that if there exists a primitive real character χ’ mod q’ st L(s,χ’) has a real exceptional β’>1-δ/logT, then we get a great bound for real part of all other zeros with |γ|<T
Specifically, if the above conditions hold then:
β=s theorem gives a certain sum less than 1, but this sum counts exactly the number of zeros not satisfying the above conditions, so none do.
How might we go about proving Linnik’s theorem?
We can show Bombieri’s theorem implies Linnik’s, so we should just try to prove this. This is still very tricky, but manageable
Let χmod q be induced by χ’mod q’ and Re(s)>1/2. Then show that:
-L’/L(s,χ)=-L’/L(s,χ’) + O(logq)
Looking at the difference of these two as sums, we see this is easy
If we have χmodq, q=<T, then what can we say about the logarithm of L(σ+iv,χ) and the logarithmic derivative when σ is larger than 1-c/logT, for a c we choose specifically?
Well if σ>σ0=1+c/logT then it is easy to show that the first is less than loglogT+O(1) and the second is O(logT).
So assume that it is no between these values. By rewriting the logarithmic derivative here as the difference between two L functions we can calculate it as a sum of zeros and bound those sum to give the same bound. Then the logarithm is bounded by an integral + logarithm of the L function at σ0+iv and as the integral is O(1) we are done.
In fact, these bounds are valid for ζ and we can remove the restriction 2<|v| if there is no exceptional zero of L.
How can we use derivatives of the L function to detect zeros of L?
We can show that if L zeros at some point then there will be large derivatives near here. Formula are messy and I still have to go over the proofs! If there are multiple zeros we have to look at multiple derivatives to assure a large one. This is all set out with lots of precision in notes
What is Turans power sum method?
How do we use it to effectively give a lower bound for the derivatives of F’/F at particular points?
If we have n complex numbers, and some K>n, then there is some
K+1<K+n such that the modulus of the sum of each of these n numbers to the power k is larger than the maximum modulus of each one, divided by 32, all to the power k.
Proof is long
Use the expression for this as a sum over poles. Is then quite obvious.
