## Homework from Lecture 1

David Farmer

Problems I and II are independent, and it is better to do one of them well than both of them poorly. Everyone should do problem III.

Problem I) Numerical experimentation is a useful way to guess what is true. This can help avoid wasted work trying to prove something that isn't true, or it can suggest an answer which can then be proven.

For each function below, determine if $$F_j(T) := \int_0^T f_j(t) dt$$ has a main term, or whether it is 0 on average and the remainder is just noise. If you think there is a main term, then subtract it off and try to guess whether there is a secondary main term.

1. $f_1(t) = Re \zeta(\frac12 + i t)$
2. $f_2(t) = Im \zeta(\frac12 + i t)$
3. $f_3(t) = Z(t)$,   the Hardy Z-function.
Lessons to be learned in this exercise:
1. Sometimes the built-in integration methods in a computer algebra system are useful, and sometimes they are not. In this excercise they are not, but don't just take my word for it.
2. If your method of computing $F_j(T)$ requires re-evaluating $f_j(t)$ for many $0 < t < T$, every time you plug in a new $T$, then the range of $T$ you can explore will be severly limited.
3. You know that you are going to have to do almost the same calculation three times. Set things up so that it takes you almost no effort to do the second and third cases.
4. The first step is to build intuition, probably by making a plot. A plot looks the same whether the function is accurate to 3 digits or 30 digits.
After you complete the exercise, look up Selberg's theorem on the distribution of $\zeta(\frac12 + i t)$, and look at Theorem 1 in "On the integral of Hardy's function Z(t)" by M. A. Korolev. Are your observations consistent with those theorems?

Problem II) Plot the Hardy Z-function on several intervals. Look carefully at the plots, and try to gain an intuition for the relationship between the spacing between the zeros and the shape of the graph.

Now we want to determine to what extent the Z-function is really just a function of its zeros. So we want to make another function with the same zeros (in a particular range) as the Z-function, and then see if the two functions look the same.

Choose an interval, find all the zeros of the zeta function in that interval, and then make another function with those same zeros.

The easiest thing to do is to make a polynomial with those zeros. Does the polynomial look like the Z-function? (My guess is that your answer will be 'no'.)

How can we make a function which does a beter job than a polynomial? One possibility is to start with a function that already has a lot of zeros and then "move around" the zeros to be where you want them. The function $\cos(K t)$ is a good choice, provided that you choose the parameter $K$ properly.

Make such a function, and see if it looks like the Z-function. (My guess is that the answer will be 'yes'). But even if it does look like the Z-function, it probably will be off be a constant factor. This is analogous to having two polynomials with the same zeros: one will be a constant multple of the other. Determine the constant factor for your example.

Now repeat the process in another region. And then another. Hopefully you have automated the process so that this is easy. Do you see any pattern in the multiplicative constant, as a function of where you base your model?

Problem III) Before Tuesday morning, read: Sections 3.2 and 3.6 of "Computational methods and experiments in analytic number theory" by Michael Rubinstein. Only read other sections as needed to understand the assigned sections. The paper is available at http://front.math.ucdavis.edu/0412.5181.