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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – November 2008

    MT 3805 – ANALYTIC NUMBER THEORY

 

 

 

Date : 07-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

 

ANSWER ALL QUESTIONS

 

I    a)  Prove that if the integer  has r distinct odd prime factors , then

[OR]

1. b) Prove that the identity function is completely multiplicative.                                        (5)

 

1. c) i) Write the relation between  and .

.     ii) Let the arithmetic function be multiplicative. Then   prove that is completely

multiplicative if, and only if  for all .                 .                   (7 + 8)

[OR]

1. d) Prove that the set of all arithmetic functions f  with f(1)0 forms an abelian group

under Dirichlet product                                                                                                                  (15)

 

 

 

II   a)  Write a note on the test for divisibility by 11.

[OR]

1. b) Write a product formula for .                                                                                                   (5)

 

1. c) If f has a continuous derivative on the interval    .             .                                             (15)

[OR]

1. d) If   prove that
2. i) .
3. ii)                            (7 +  8)

 

 

 

 

 

 

 

III   a) If , then prove that the linear congruence  has exactly one

solution.

[OR]

1. b) State and prove Wolstenholme’s                                                                                        (5)

 

1. c) i)  State and prove Lagrange’s theorem for polynomial congruences.
2.         ii)  Let  . Prove that  is composite.                                                              (7 + 8)

[OR]

1. d) i) Solve the congruence (mod 120).
2. ii) Write any two properties of residue classes.                             (7 + 8)

 

 

IV   a) Write a note on  quadratic residues and give an example.

[OR]

1. b) Prove that Legendre’s symbol is a completely multiplicative function.                             (5)

 

1. c) State and prove Gauss’ lemma. Also derive the value of m defined in Gauss

lemma.

[OR]

1. d) Determine those odd primes p for which 3 is a quadratic residue and those for

which it is a nonresidue.                                                                                                                (15)

 

 

V    a) State and prove  reciprocity law.

[OR]

1. b) Write a note on partitions.                                                                                      (5)

 

1. c) Derive a generating function for

[OR]

1. d) State and prove Euler’s pentagonal-number theorem                             (15)

 

 

 

 

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