By Edward Burger

2 DVD set with 24 lectures half-hour each one for a complete of 720 minutes...Performers: Taught through: Professor Edward B. Burger, Williams College.Annotation Lectures 1-12 of 24."Course No. 1495"Lecture 1. quantity concept and mathematical learn -- lecture 2. average numbers and their personalities -- lecture three. Triangular numbers and their progressions -- lecture four. Geometric progressions, exponential progress -- lecture five. Recurrence sequences -- lecture 6. The Binet formulation and towers of Hanoi -- lecture 7. The classical thought of major numbers -- lecture eight. Euler's product formulation and divisibility -- lecture nine. The major quantity theorem and Riemann -- lecture 10. department set of rules and modular mathematics -- lecture eleven. Cryptography and Fermat's little theorem -- lecture 12. The RSA encryption scheme.Summary Professor Burger starts with an summary of the high-level strategies. subsequent, he offers a step by step rationalization of the formulation and calculations that lay on the middle of every conundrum. via transparent causes, exciting anecdotes, and enlightening demonstrations, Professor Burger makes this exciting box of research obtainable for an individual who appreciates the interesting nature of numbers. -- writer.

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**Extra resources for An Introduction to Number Theory (Guidebook, parts 1,2)**

**Example text**

3 for the definition of ζ(s) when s ∈ R and s > 1). Elementary proofs were discovered around 1949 by A. Selberg and P. Erd¨os. Their proofs do not involve ζ(s) and complex function theory, hence the name “elementary”. There are other elementary proofs of the prime number theorem since the appearance of the work of Selberg and Erd¨os, one of which is [4]. The 41 February 13, 2009 16:7 World Scientific Book - 9in x 6in 42 AnalyticalNumberTheory Analytic Number Theory for Undergraduates proof given in [4] relies on proving an equivalent statement of the Prime Number Theorem and the mean value of µ(n).

4. For real number x ≥ 1, let θ(x) = ln p. 3. For real number x ≥ 1, we have √ θ(x) = ψ(x) + O( x). Proof. We first note that the difference of ψ(x) and θ(x) is ψ(x) − θ(x) = ln p pm ≤x m≤2 ln p + = √ p≤ x m=2 1. 1). 3, we deduce the following corollary. 4. For x ≥ 4, there exist real positive constants c1 and c2 such that c1 x ≤ θ(x) ≤ c2 x. 1. 5. For each positive real x ≥ 4, c2 x c1 x . ≤ π(x) ≤ ln x ln x Proof. 3. 4, A(t) = n≤t a(n) = θ(t) ≪ t. 8) is 1 1 − θ(x) ln x ln x x − 1 1 − ln t ln x θ(t) 2 x = 2 √ x θ(t) dt ≪ t ln2 t x 2 ′ dt dt ln2 t x dt ln2 t 2 x √ x dt ≪ 2 .

3. For real number x ≥ 1, we have √ θ(x) = ψ(x) + O( x). Proof. We first note that the difference of ψ(x) and θ(x) is ψ(x) − θ(x) = ln p pm ≤x m≤2 ln p + = √ p≤ x m=2 1. 1). 3, we deduce the following corollary. 4. For x ≥ 4, there exist real positive constants c1 and c2 such that c1 x ≤ θ(x) ≤ c2 x. 1. 5. For each positive real x ≥ 4, c2 x c1 x . ≤ π(x) ≤ ln x ln x Proof. 3. 4, A(t) = n≤t a(n) = θ(t) ≪ t. 8) is 1 1 − θ(x) ln x ln x x − 1 1 − ln t ln x θ(t) 2 x = 2 √ x θ(t) dt ≪ t ln2 t x 2 ′ dt dt ln2 t x dt ln2 t 2 x √ x dt ≪ 2 .