Math 675: Syllabus Introduction to Number Theory

Math 675: Introduction to Analytic Number Theory: Syllabus

Text (required): H. Davenport, Multiplicative Number Theory, Second edition, Revised by H. L. Montgomery, Springer-Verlag: New York 1980

Text (optional): H. L. Montgomery and R. C. Vaughn, Multiplicative Number Theory: I. Classical Theory, Cambridge University Press, Cambridge 2007.

Text (optional): G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge 1995.

Text (optional): H. Iwaniec and E. Kowalski, Analytic Number Theory, Colloquium Publications Vol. 53, American Mathematical Society: Providence, RI 2004.

(Version December 14, 2011)

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# date Refs: Sections in: Davenport[D]; Montgomery-Vaughan [MV],Tenenbaum [T] HW due dates problems

0

1 9/7 Introduction: 0.1 Addition vs Multiplication, Math Logic, 0.2 Distribution of Primes

2 9/9 0.3 Probabilistic Number Theory; 1.1 Chebyshev estimates, Upper Bound [T, 1.2]

3 9/12 1.1 Nair lower bound [T. 1.2, 1.3] 1.2 Recap: Gelfond-Shnirelman approach to lower bounds, 1.3 Tauberian theorems

4 9/14 A.1 Riemann-Stieltjes Integrals; Partial Summation [M, Appendix A]

5 9/16 1.3 p-Adic vaulation of n!, 1.4 Mertens's First Theorem for (log p)/p [T. 1.3, 1.4]

6 9/19 1.5 Asymptotic formulas for \sum 1/p, Prod(1-1/p) [T. 1.5]; 1.5B. Sieves vs. Mertens's Formula; 1.6 Chebyshev limits [T. 1.7] HW#1 due

7 9/21 A.2 Bernoulli numbers and functions; Euler-Maclaurin Summation; Harmonic Series [T, 0.2]

8 9/23 2.1 Arithmetic Functions; [T. 2.1, 2.2] 2.2 Ring of Arithmetic Functions; Formal Dirichlet Series [T. 2.3, 2.4]
9 9/26 2.3 Mobius inversion formula: two forms [T. 2.5]
10 9/28 2.4 Euler totient function [T. 2.7] 3.1 Average order- arithmetic fns [T. 3.1] 3.2 Divisor function-average order-Hyperbola method [T.3.2]

11 9/30 3.3 Euler Totient Fn-Average Order; Farey Sequence [T. 3.4] ; 3.4 Mobius function-Average order-PNT equivalent [T. 3.6]; fire drill HW#2 due
12 10/3 3.5 Squarefree numbers-Average Order [T. 3.7], 3.6 Multiplicative function-mean value [T 3.8]

13 10/5 3.6 Multiplicative function-mean value (Erdos- alpha-beta method)[T 3.8]

14 10/7 4.1 Sieve of Eratosthenes; Inclusion-Exclusion [T. 4.1] 4.2 Brun's Combinatorial Sieve [T. 4.2]

15 10/10 4.2 Brun's Combinatorial Sieve,[T. 4.2] 4.3 Application:Estimating Rough Numbers (parameter method)[T. 4.2] HW#3 due

16 10/12 4.3 Application:Estimating Rough Numbers [T.4.2] 4.4 Application: Upper Bound Twin Primes (statement) [T. 4.3]

17 10/14 4.4 Application: Upper Bound Twin Primes,[T.4.3] 4.5 Fundamental Lemma of Combinatorial Sieve [T. 4.2]
10/17 Fall Study Break
18 10/19 5.1 General Dirichlet Series: Absolute Convergence, 5.2 General Dirichlet Series: Conditional Convergence

19 10/21 5.2 (cont.), 5.3 General Dirichlet Series: Formulae for Abscissa of Conditional/Absolute Convergence

20 10/24 5.4 Uniqueness Theorem for General Dirichlet Series; 6.1 Dirichlet's Theorem for Primes in Arithmetic Progession (Overview) [D,Chap 1] HW#4 due
21 10/26 6.2 Dirichlet Characters; [D. Chap. 1,and 4] 6.3 Dirichlet theoreom L(1, \chi) not zero, \chi complex [D, Chap. 1]

22 10/28 6.3 L(1, \chi) not zero, \chi complex; 6.4 L(1, \chi) not zero, \chi real, m=q odd prime [D., Chap. 1]

23 10/31 6.4 (continued), 6.5 Gauss sum-I, 6.6 Proof L(1, \chi) not zero, \chi real, m=q prime [Chap. 1]
24 11/2 Class cancelled, building power failure

25 11/4 6.6 (concluded); 6.7 Evaluation of Gauss sums, sign of sum [D.-chap 2] [Stephen DeB.]

26 11/7 6.8 Goblins and Gaussians; 6.9 Cyclotomy formulas, 6.10 Application: Polygon constructions[D., Chap. 3] HW#5 due
27 11/9 6. 11 L(1, \chi) nonzero, \chi real, general modulus m; 6.12 Dirichlet formula for L(s, \chi)-Fekete polynomials [D., Chap. 4]

28 11/11 7.1 Primitive characters; 7.2 Real Primitive Characters : Kronecker symbols [D., Chap. 5]

29 11/14 7.3 Class numbers of quadratic forms 7.4 Dirichlet Class Number Formula [D. Chapter 6] [HLM lecture]

30 11/16 7.4 Dirichlet class number formula, Pell equation, 8.0 Distribution of Primes; Chebyshev revisited [D. Chap. 6, 7][HLM lecture]

31 11/18 Class Postponed

32 11/21 8.1 Riemann's memoir, 8.2 Analytic continution and functional equation via theta function [D. Chap. 8] HW#6 due

33 11/23 8.3 Functional equation for theta function, 8.4 Zeta function for quadratic fields [D. Chap. 8]

11/24 Thanksgiving

34 11/28 9.1 Functional equation for Dirichlet L-functions; 9.2 Primitive characters and Gauss sums [D, Chap. 9]

35 11/30 9.3 Character theta function functional equation [D, Chap. 9]

36 12/2 10.1 Gamma Function, 11.1 Entire functions of finite order, [D., Chap. 10, 11]

37 12/5 11.1 Entire functions of Finite order; Jensen's formula, 11.2 Hadamard product for entire functions of order 1 [D. Chap. 11]

38 12/7 11.2 (cont'd), 11.3 Paley-Wiener Theorem, 12.1 Infinite Product for xi(s), 12.2 Logarithmic derivative of xi(s) [D. Chap. 11, 12]

39 12/9 12.3 Constants A, B in Hadamard product for xi(s), 13.1 zeta(s) has no zeros on Re(s)=1, 13.2 Zero-free region for zeta(s) [D, Chap.12, 13]

40 12/12 15.0 Argument principle (Review); 15.1 Riemann formula for N(T): proof, 15.2 Estimate S(T) = O(log T). [D., Chap. 15] HW#7 due
41 12/14 17.1 Explicit formula for \psi(x), 17.2 Inverse Mellin transform of Heaviside function; 17.3 Proof of Truncated Explicit Formula; 18.1 Prime Number Theorem- \psi(x) form [D. Chap. 17, 18.1]

#	date	Refs: Sections in: Davenport[D]; Montgomery-Vaughan [MV],Tenenbaum [T]	HW due dates	problems
0
1	9/7	Introduction: 0.1 Addition vs Multiplication, Math Logic, 0.2 Distribution of Primes
2	9/9	0.3 Probabilistic Number Theory; 1.1 Chebyshev estimates, Upper Bound [T, 1.2]
3	9/12	1.1 Nair lower bound [T. 1.2, 1.3] 1.2 Recap: Gelfond-Shnirelman approach to lower bounds, 1.3 Tauberian theorems
4	9/14	A.1 Riemann-Stieltjes Integrals; Partial Summation [M, Appendix A]
5	9/16	1.3 p-Adic vaulation of n!, 1.4 Mertens's First Theorem for (log p)/p [T. 1.3, 1.4]
6	9/19	1.5 Asymptotic formulas for \sum 1/p, Prod(1-1/p) [T. 1.5]; 1.5B. Sieves vs. Mertens's Formula; 1.6 Chebyshev limits [T. 1.7]	HW#1 due
7	9/21	A.2 Bernoulli numbers and functions; Euler-Maclaurin Summation; Harmonic Series [T, 0.2]
8	9/23	2.1 Arithmetic Functions; [T. 2.1, 2.2] 2.2 Ring of Arithmetic Functions; Formal Dirichlet Series [T. 2.3, 2.4]
9	9/26	2.3 Mobius inversion formula: two forms [T. 2.5]
10	9/28	2.4 Euler totient function [T. 2.7] 3.1 Average order- arithmetic fns [T. 3.1] 3.2 Divisor function-average order-Hyperbola method [T.3.2]
11	9/30	3.3 Euler Totient Fn-Average Order; Farey Sequence [T. 3.4] ; 3.4 Mobius function-Average order-PNT equivalent [T. 3.6]; fire drill	HW#2 due
12	10/3	3.5 Squarefree numbers-Average Order [T. 3.7], 3.6 Multiplicative function-mean value [T 3.8]
13	10/5	3.6 Multiplicative function-mean value (Erdos- alpha-beta method)[T 3.8]
14	10/7	4.1 Sieve of Eratosthenes; Inclusion-Exclusion [T. 4.1] 4.2 Brun's Combinatorial Sieve [T. 4.2]
15	10/10	4.2 Brun's Combinatorial Sieve,[T. 4.2] 4.3 Application:Estimating Rough Numbers (parameter method)[T. 4.2]	HW#3 due
16	10/12	4.3 Application:Estimating Rough Numbers [T.4.2] 4.4 Application: Upper Bound Twin Primes (statement) [T. 4.3]
17	10/14	4.4 Application: Upper Bound Twin Primes,[T.4.3] 4.5 Fundamental Lemma of Combinatorial Sieve [T. 4.2]
	10/17	Fall Study Break
18	10/19	5.1 General Dirichlet Series: Absolute Convergence, 5.2 General Dirichlet Series: Conditional Convergence
19	10/21	5.2 (cont.), 5.3 General Dirichlet Series: Formulae for Abscissa of Conditional/Absolute Convergence
20	10/24	5.4 Uniqueness Theorem for General Dirichlet Series; 6.1 Dirichlet's Theorem for Primes in Arithmetic Progession (Overview) [D,Chap 1]	HW#4 due
21	10/26	6.2 Dirichlet Characters; [D. Chap. 1,and 4] 6.3 Dirichlet theoreom L(1, \chi) not zero, \chi complex [D, Chap. 1]
22	10/28	6.3 L(1, \chi) not zero, \chi complex; 6.4 L(1, \chi) not zero, \chi real, m=q odd prime [D., Chap. 1]
23	10/31	6.4 (continued), 6.5 Gauss sum-I, 6.6 Proof L(1, \chi) not zero, \chi real, m=q prime [Chap. 1]
24	11/2	Class cancelled, building power failure
25	11/4	6.6 (concluded); 6.7 Evaluation of Gauss sums, sign of sum [D.-chap 2] [Stephen DeB.]
26	11/7	6.8 Goblins and Gaussians; 6.9 Cyclotomy formulas, 6.10 Application: Polygon constructions[D., Chap. 3]	HW#5 due
27	11/9	6. 11 L(1, \chi) nonzero, \chi real, general modulus m; 6.12 Dirichlet formula for L(s, \chi)-Fekete polynomials [D., Chap. 4]
28	11/11	7.1 Primitive characters; 7.2 Real Primitive Characters : Kronecker symbols [D., Chap. 5]
29	11/14	7.3 Class numbers of quadratic forms 7.4 Dirichlet Class Number Formula [D. Chapter 6] [HLM lecture]
30	11/16	7.4 Dirichlet class number formula, Pell equation, 8.0 Distribution of Primes; Chebyshev revisited [D. Chap. 6, 7][HLM lecture]
31	11/18	Class Postponed
32	11/21	8.1 Riemann's memoir, 8.2 Analytic continution and functional equation via theta function [D. Chap. 8]	HW#6 due
33	11/23	8.3 Functional equation for theta function, 8.4 Zeta function for quadratic fields [D. Chap. 8]
	11/24	Thanksgiving
34	11/28	9.1 Functional equation for Dirichlet L-functions; 9.2 Primitive characters and Gauss sums [D, Chap. 9]
35	11/30	9.3 Character theta function functional equation [D, Chap. 9]
36	12/2	10.1 Gamma Function, 11.1 Entire functions of finite order, [D., Chap. 10, 11]
37	12/5	11.1 Entire functions of Finite order; Jensen's formula, 11.2 Hadamard product for entire functions of order 1 [D. Chap. 11]
38	12/7	11.2 (cont'd), 11.3 Paley-Wiener Theorem, 12.1 Infinite Product for xi(s), 12.2 Logarithmic derivative of xi(s) [D. Chap. 11, 12]
39	12/9	12.3 Constants A, B in Hadamard product for xi(s), 13.1 zeta(s) has no zeros on Re(s)=1, 13.2 Zero-free region for zeta(s) [D, Chap.12, 13]
40	12/12	15.0 Argument principle (Review); 15.1 Riemann formula for N(T): proof, 15.2 Estimate S(T) = O(log T). [D., Chap. 15]	HW#7 due
41	12/14	17.1 Explicit formula for \psi(x), 17.2 Inverse Mellin transform of Heaviside function; 17.3 Proof of Truncated Explicit Formula; 18.1 Prime Number Theorem- \psi(x) form [D. Chap. 17, 18.1]