Zero Free Region

Question 1

State the classical zero free region for the ζ function

And for L functions of primitive complex characters χmodq.

Answer 1

We know that any zero has to lie in the critical strip anyway. So suppose the zeta function has a zero z=b+it. Then there is a constant c>0 such that:
b<=1-c/[(log(|t|+2))+logq]

Question 2

How does the classical zero free region of L functions change if χ mod q is a real primitive character?

Answer 2

It doesn’t bit is still

b0 then c is now a function of δ and the zero free region holds for all roots z=b+it with |t|>δ/logq

Question 3

Sketch a proof of the classical zero free region for the ζ function.

Answer 3

Consider ζ with ζ(ρ)=0,ρ=β+γi. Then as ζ has a pole at 1, we can assume that the imaginary part is larger than c for some small constant. Now consider the real part of certain logarithmic derivatives of the zeta function of some specific values and use a handy trigonometric identity to show this sum is larger than zero, where this sum includes things of the form:
ζ’/ζ(σ), ζ’/ζ(σ+γi), ζ’/ζ(σ+2iγ)
With some real σ>1.

Then use an existing simple bound for the first expression, and then use our important lemma to express the logarithmic derivatives in terms of zeros near ρ. Then by dropping all terms but the one with ρ in, get a bound for the other two terms in terms of the β part of the root. Then put it together, and choose a specific value of σ>1 to make it work correctly.

Question 4

Show, with specifics, that for a primitive quadratic character mod q there is at most one zero near the real line.

Answer 4

There is a d>0 such that for all q and all primitive quadratic characters, L has at most one zero in the region Re(s)>1-d/logq,
|Im(s)|1 and write it using our lemma as an expression with a sum over a set of zeros of the completed L function. Then we can discard all these summands except the ones for our two roots here, and we get an upper bound for the logarithmic derivative. However, as σ is real we get another simple expression for this logarithmic derivative, given by:
-L’/L(σ,χ)=Ο(1)-1/(σ-1)

Then use a specific value of σ to finish the proof.

Question 5

What is Siegel's theorem for primitive character χmodq?
Where is the theorem defective, as in it becomes impossible for us to use the proof to calculate the constants that exist?
Use the class number formula to give info about the class number from this formula

Answer 5

Let ε>0. Then there exists c(ε)=c such that L(1,χ)>c•q^-ε
Also, if L(β,χ)=0 then
β=

Well just plug in this info, the info from the class number formula, and the function equation for L.

Question 6

Give a bound for the L function and its derivative for any non principal character at some σ such that it is sufficiently close to 1:

Answer 6

If 1-1/logq<1 then by using the stieljes integral at the start of the course and the sum formula for L, an integral which we can do by parts to get a simpler integral. We can then just bound the sum part of the integral using periodic out of the character sum. Then computing the integral directly gives the answer.

Question 7

Show that the two statements of Siegel’s theorem are equivalent.

Answer 7

For 1) implies 2) use the mean value theorem and our bound for L’ and L
For 2) implies 1) rewrite log(L(1,χ)) as an integral and then something else. Then bound the integral and bound the other term by comparison with ζ

Question 8

Prove the following theorem:
Suppose χ1 & χ2 are distinct primitive characters with conductors both less than q. Then there exists a constant d>0 such that at most on of the two L functions has a real zero β with
β>1-d/logq

Answer 8

Suppose they both have zeros
βi>1-d/logq. Consider the function F(s) which is the product of ζ and the L functions of χ1, χ2, & χ1•χ2.
Take σ>1 and evaluate -F’/F at σ. Then this is larger then 0, by factoring appropriately.
Then bound -ζ’/ζ(σ) as usual, and by throwing away zeros in our estimate of -L’/L(s,χi) get the bound:
-L’/L(s,χi)=/L(s,χ1•χ2).
Now put all these together, and set σ at the appropriate value (recalling the bound on βi here) to show that for small enough d this cannot be true.

Question 9

Pull together all we have about zeros of primitive L- functions.

Answer 9

THEOREM:
There is a d>0 such that for all primitive character mod q, the corresponding L function has no zero in the region:
{σ+it : σ > 1-d/(logq+log(|t|+2))}
With at most one exception. Such an exception is necessarily real and can only occur for a real primitive character. Moreover, among all real primitive characters χ’modq’ for q’1-d/logq. Finally, if such a zero exists, it satisfies, for each ε>0:
1-β>=Cε•q^-ε

Question 10

What is the Perron summation formula?

Can you briefly sketch a proof?

Answer 10

Σχ(p^n)•logp=1/(2πi)•INTEGRAL(-L’/L(s,χ)•x^s•ds/s) + O(x(logx)^2/T)

Where the integral is between σ+iT and σ-iT, and the sum is over all p^n<x

Note. This estimate is also valid I we replace χ(p^n) by 1, and L by ζ.

Hard… Look at book