Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Sums and Differences
- Chapter 2 Products and Divisibility
- Chapter 3 Order of Magnitude
- Chapter 4 Averages
- Interlude 1 Calculus
- Chapter 5 Primes
- Interlude 2 Series
- Chapter 6 Basel Problem
- Chapter 7 Euler's Product
- Interlude 3 Complex Numbers
- Chapter 8 The Riemann Zeta Function
- Chapter 9 Symmetry
- Chapter 10 Explicit Formula
- Interlude 4 Modular Arithmetic
- Chapter 11 Pell's Equation
- Chapter 12 Elliptic Curves
- Chapter 13 Analytic Theory of Algebraic Numbers
- Solutions
- Bibliography
- Index
Interlude 1 - Calculus
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Preface
- Chapter 1 Sums and Differences
- Chapter 2 Products and Divisibility
- Chapter 3 Order of Magnitude
- Chapter 4 Averages
- Interlude 1 Calculus
- Chapter 5 Primes
- Interlude 2 Series
- Chapter 6 Basel Problem
- Chapter 7 Euler's Product
- Interlude 3 Complex Numbers
- Chapter 8 The Riemann Zeta Function
- Chapter 9 Symmetry
- Chapter 10 Explicit Formula
- Interlude 4 Modular Arithmetic
- Chapter 11 Pell's Equation
- Chapter 12 Elliptic Curves
- Chapter 13 Analytic Theory of Algebraic Numbers
- Solutions
- Bibliography
- Index
Summary
The techniques discussed in the previous chapters can be pushed a little further, at the cost of a lot of work. To make real progress, however, we need to study the prime numbers themselves. How are the primes distributed among the integers? Is there any pattern? This is a very deep question, which was alluded to at the beginning of Chapter 3. This Interlude makes a detour away from number theory to explain the ideas from calculus that we will need. It covers things I wish you had learned but, based on my experience, I expect you did not. I can't force you to read it, but if you skip it, please refer back to it later.
Linear Approximations
Although you might not notice, all of differential calculus is about a single idea: Complicated functions can often be approximated, on a small scale anyway, by straight lines. What good is such an approximation? Many textbooks will have a (rather unconvincing) application, something like “approximate the square root of 1.037.” In fact, almost everything that happens in calculus is an application of this idea.
For example, one learns that the graph of function y = f(x) increases at point x = a if the derivative f′(a) is positive. Why is this true? It's because of the linear approximation idea: The graph increases if the straight line approximating the graph increases. For a line, it's easy to see that it is increasing if the slope is positive. That slope is f′(a).
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- Chapter
- Information
- A Primer of Analytic Number TheoryFrom Pythagoras to Riemann, pp. 83 - 95Publisher: Cambridge University PressPrint publication year: 2003