What is a function?

George E. Hrabovsky, President, MAST

Where We Have Been

Last time I wrote down the Riemann hypothesis (RH) and asked the first twenty four questions of this project. The first of these questions is, "What is a function?"  That will be our starting point for this session. These are the foundational questions for this project:

    1. What, exactly, is a function?

    2. What is a zeta function?

    3. What is the Riemann zeta function?

    4. What is the zero of a function?

    5. What is a nontrivial zero?

    6. What is meant by Re( s )?

    7. What does mean?

    8. What does Li( x ) mean?

    9. What does O mean?

    10. What does mean?

    11. What does mean?

    12. What does µ( x ) mean?

    13. What does mean?

    14. What is a congruence?

    15. What is an algebraic function?

    16. What is a function field?

    17. What is an algebraic curve?

    18. What does P ( u ) mean?

    19. What does mean?

    20. What does mean?

    21. What does mean?

    22. What is a variety?

    23. What is an algebraic variety?

    24. What is a finite field?

Session 2: What is a function?

Last time I introduced an important source for this project [1]. I will now introduce a second source, Eric W. Weisstein's, "CRC Concise Encyclopedia of Mathematics" [2]. There is a second edition that I do not yet have. You can also find an online version at the Wolfram Research web site called "Eric Weisstein's World of Mathematics" at www.wolfram.com. I looked up functions and came up with this explanation: a function is some rule that associates a unique object to every element of a set. Given a pair of sets, and , a function from to is a rule such that for every element of there exists a unique element of such that the element of is a determined by the rule operating on the element of . We can write all of this in mathematical shorthand,

where, means, "for all," means, "an element of," means, "the function from the set to the set ," and means, "there exists a unique something."  To understand this in a more concrete way, we need to see some examples. One rule that comes immediately to mind is the square of a number. Let us say that we have the set of the first four natural numbers as set ,

We state that the set is the set of all natural numbers. This set is given the special symbol N. We define the rule as the square of some element,

where whatever element we choose fills in the place of the little square.  In our case, we have,

The results is,

This definition leads to the question:

    25. What is a set?

What is a set?

A set is a collection of objects called elements. If we wish to write that is an element of the set , we note from the section above .  We have successfully answered the first foundational question.

What is a -function?

This is the second foundational question. According to reference 2, a zeta function satisfies certain properties and is computed as an infinite sum of negative powers.

This is not very satisfactory.  Returning to reference 1, we learn the properties that a zeta function satisfies:

    1. They are meromorphic on the complex plane.

    2. They have Dirichlet series expansions.

    3. They have Euler product expansions.

    4. They satisfy certain functional equations.

These definitions combine to raise further questions:

    26. What is an infinite sum?

    27. What does it mean to be meromorphic?

    28. What is the complex plane?

    29. What is a Dirichlet series?

    30. What is a series expansion?

    31. What is an Euler product?

    32. What is a product expansion?

    33. What is a functional equation?

What is an infinite sum?

I was unable to come up with a good definition of an infinite sum in references 1 or 2.  Instead I looked up sums in reference 2.  As you might expect I found that a sum is the result of addition.  There is a short-hand way of writing a sum.  Instead of 1 + 2 + 3 + 4 = 10, we have a sum of four terms,

This is called the sigma notation .  An nth sum would be written

and an infinite sum would be

There is a way of calculating the sum called the Euler-Maclaurin Integration formula:

We have many more questions, now.

    34. What does infinite mean?

    35. What is integration?

    36. What does mean?

    37. What does mean?

    38. What does 2! mean?

    39. What does mean?

    40. What does mean?

Book Review: The CRC Concise Encyclopedia of Mathematics

Eric W. Waistline, "CRC Concise Encyclopedia of Mathematics ," CRC Press (1999).

This is a single large volume that has a very special place at my desk. This is one of the finest mathematical references in print, as far as I am concerned. Not only does it have short articles that provide good information on almost everything in mathematics, it is also very well cross-referenced. It is a very large book, but don't be intimidated; you do not need a high degree of mathematical sophistication to use this. Make no mistake, this is a reference to be used. At around a $100, it is a very good buy.

You need not worry about it becoming out-of-date either. There is an online version where articles are always being updated, and new ones are being included, too. I highly recommend it.

References

1. Kayos Ito, "Encyclopedic Dictionary of Mathematics," MIT Press (Paperback Second Edition), (1993).

2. Eric W. Waistline, "CRC Concise Encyclopedia of Mathematics," CRC Press (1999).

Created by Mathematica  (November 4, 2004).