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Transcript

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.