LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 25

THIRD SEMESTER – NOV 2006

         MT 3805 – ANALYTIC NUMBER THEORY

 

 

Date & Time : 30-10-2006/9.00-12.00      Dept. No.                                                       Max. : 100 Marks

 

 

 

 

Answer ALL questions.

1. a) i) Define Mobius function and Euler function
2. ii) Prove that for n≥1.                                                     (2+3)

Or

iii) Prove that log n = and.    (5)

1. b) i) Prove that the set of all arithmetical functions f with f(1)≠0 forms an abelian group

with respect to Dirichlet product , the identity element being the function I.

Or

1. ii) Let f be multiplicative. Then prove that f is completely multiplicative if and only if

f for all n1.

iii) If f is multiplicative then prove that.        (10+5)

1. a) i) State and prove Euler’s summation formula.

Or

1. ii) Prove that where C is Euler’s constant. (5)
2. b) i) State and prove weak and strong versions of Dirichlet asymptotic formulae for

the partial sums of the divisor function d(n).

Or

1. ii) ) State and prove Asymptotic formulae for the partial sums of divisor functions

and                                                             (15)

III. a) i)  An integer n>0 is divisible by 9 if and only if the sum of its digits in its decimal

expansion is divisible by 9. Prove this using congruences.

Or

1. ii) If acand if d= (m,c), then prove that a≡b. (5)
2. b) i) State and prove Lagrange’s theorem.
3. ii) For any prime p prove that all the coefficients of the polynomial

f(x)=(x-1)(x-2)(x-3)…………(x-p+1)-x+1 are divisible by p.       (10+5)

Or

iii) If (a,m)=1, prove that  the solution of the linear congruence ax≡b (mod m) is

is given by x≡ba (mod m).

1. iv) State and prove Chinese remainder theorem. (6+9)

 

 

 

 

 

 

1. a) i) Let p be an odd prime. Then for all n prove that.

Or

1. ii) Prove that Legendre’s symbol () is a completely multiplicative

function of n.                                                                                     (5)

1. b) i) For every odd prime p, Prove that and

.

1. ii) State and prove Gauss’ Lemma. (7+8)

Or

iii) State and prove Quadratic reciprocity law. Use it  to determine those odd

primes p for which 3 is a quadratic residue and those for which it is a

nonresidue                                                                                      (15)

1. a) i) Evaluate where P is an odd positive integer.

Or

1999. ii) Determine whether 888 is a quadratic residue or nonresidue of the prime 1999.
2000. b) i) Prove that for <1 ,,where p(0)=1 and

p(n) is the partition function.

Or

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

 

 

 

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AB 34

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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