Download Homework and Senior Projects 11

Chapter 8
Complex Numbers
Homework
1) Find k  2 such that M(k) = 0 and prove it.
2) Calculate the Farey sequence F6
3) Find D(5), the sum of the differences between the Farey sequence of 5 and each
1/A(i).
4) A complex number is said to be in trigonometric form when written as follows:
C = r(cos() + isin())
Prove that the product of two complex numbers is of the form:
rs(cos(1+2) + isin(1+2))
5) In a couple of paragraphs, describe the process of analytic continuation as it
applies to Riemann’s extended zeta function, as well as an example of an analytic
continuation and a brief explanation of a germ.
Senior Projects
1. Explore why sqrt(n) is the largest number you'll have to go to to find all of the factors
of a number.
2. Explore the proof that for any s, the value of ζ(s) is equal to the infinite product over
the primes of
numbers.
. Consider what this means in terms of the prime
3. Develop a computational approach for verifying the Riemann Hypothesis.