AnalyticNumbertheoryclass3

Reminder Final class Wednesday27th ZoeDeadline Sat 23rd

A13387W1

SECOND PUBLIC EXAMINATION

Honour School of Mathematics Part C: Paper C3.8Honour School of Mathematics and Computer Science Part C: Paper C3.8

Honour School of Mathematics and Philosophy Part C: Paper C3.8

ANALYTIC NUMBER THEORY

Trinity Term 2018

MONDAY, 4 JUNE 2018, 2.30pm to 4.15pm

You may submit answers to as many questions as you wish but only the best two will count forthe total mark. All questions are worth 25 marks.

You should ensure that you:

• start a new answer booklet for each question which you attempt.

• indicate on the front page of the answer booklet which question you have attempted in thatbooklet.

• cross out all rough working and any working you do not want to be marked. If you have usedseparate answer booklets for rough work please cross through the front of each such bookletand attach these answer booklets at the back of your work.

• hand in your answers in numerical order.

If you do not attempt any questions, you should still hand in an answer booklet with the frontsheet completed.

Do not turn this page until you are told that you may do so

Page 1 of 3

3. (a) [2 marks] Show that

⇣(�) 6 1 +1

� � 1

for all real � > 1.

(b) [5 marks] Show that if <s > 1 then

⇣(s)4

⇣(2s)=

1X

n=1

⌧(n)2

ns,

where ⌧(n) denotes the number of divisors of n. [You need not rigorously justify theconvergence of the series, or any rearrangements you make.]

(c) [6 marks] By choosing s = 1 + 1logX , or otherwise, show that

X

n6X

⌧(n)2 ⌧ X log4X

uniformly for X > 2.

(d) [3 marks] Let W : R ! R be smooth with compact support in [1, 10] andRRWdx = 1.

Let � > 1. Show that

X

n

⌧(n)2W (n/X) =1

2⇡i

Z �+i1

��i1

⇣(s)4

⇣(2s)W̃ (s)Xs

ds.

[Results about the Mellin transform may be stated without proof, but they should be clearlystated ]

(e) [6 marks] Show that the residue of ⇣(s)4XsW̃ (s)/⇣(2s) at s = 1 is cX log3X+O(X log2X),

where c is a constant you should specify. [You may use the fact that ⇣(2) = ⇡2/6.]

(f) [3 marks] Very briefly, and without details, explain why one expects that, as X ! 1,

X

n

⌧(n)2W (n/X) ⇠ cX log3X.

A13387W1 Page 3 of 3 End of Last Page

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A13387W1

SECOND PUBLIC EXAMINATION

Honour School of Mathematics Part C: Paper C3.8Honour School of Mathematic and Philosophy Part C: Paper C3.8Honour School of Mathematics and Statistics Part C: Paper C3.8

ANALYTIC NUMBER THEORY

Trinity Term 2017

MONDAY, 5 JUNE 2017, 9.30am to 11.15am

You may submit answers to as many questions as you wish but only the best two will count forthe total mark. All questions are worth 25 marks.

You should ensure that you:

• start a new answer booklet for each question which you attempt.

• indicate on the front page of the answer booklet which question you have attempted in thatbooklet.

• cross out all rough working and any working you do not want to be marked. If you have usedseparate answer booklets for rough work please cross through the front of each such bookletand attach these answer booklets at the back of your work.

• hand in your answers in numerical order.

If you do not attempt any questions, you should still hand in an answer booklet with the frontsheet completed.

Do not turn this page until you are told that you may do so

Page 1 of 2

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