th_02

th_02 - s . Exercise 3. Show that the equation x 1 + x 2 +...

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Number Theory 2nd Take Home due: Wednesday, April 29 All exercises are worth 5 points. Please select five problems since there is a cap of 25 points. If you decide to do more than five problems, please indicate which ones you want graded since otherwise the grader will be free to chose problems at will. Hint: Not all problems are equally hard, so finding the ones not to submit is part of the task. This is a take home. It’s just like home work, except that it is (a) comprehensive, (b) closed book and (c) to be done without collaboration; the last point includes me: I won’t answer questions on the assignment during office hours. Instead, if you feel that a problem needs clarification, send me an e-mail. I will answer to the class, so that everybody is on the same page. Recall that the Dirichlet series L ( s,χ ) for a character χ is: L ( s,χ ) := n =1 χ ( n ) n s Exercise 1. Show that L ( s,χ ) X n =1 χ ( n ) φ ( n ) n s = L ( s - 1 ) for sufficiently large s . Exercise 2. Prove or disprove that there is a Dirichlet series A ( s ) = n =1 a n n s with A ( s ) = 1 s for all sufficiently large
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Unformatted text preview: s . Exercise 3. Show that the equation x 1 + x 2 + + x j = n has n-1 j-1 solutions in positive integers. Note: solutions dier if they dier in at least one index, i.e. (1 , 1 , 2) diers from (2 , 1 , 1). Exercise 4. Show that there are 2 n-1 ways of writing n as a sum of strictly positive integers where order of summands is taken into account (i.e., 1 + 2 + 1 is dierent from 2 + 1 + 1 and both are dierent from 1 + 1 + 2). Exercise 5. Prove that the number of positive fractions 1 that have denominator n (when reduced to lowest terms) is (1) + (2) + (3) + + ( n ) . Exercise 6. Assume m ( m ) = n ( n ). Show m = n . Exercise 7. Let N be a positive integer and let f and F be arithmetic functions. Show that the following are equivalent: 1. F ( n ) = b N/n c k =1 f ( kn ) for all n . 2. f ( n ) = b N/n c k =1 ( k ) F ( nk ) for all n . (Here is the Mobius function.)...
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