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Page 51

over all integers, diverges. This suggests a way of describing the size of a set S of

integers. Let us say that S is "small" if 21/n, summed over S, converges, and

otherwise that S is "large." The

Can we say that a

either too few or too many zeros in its decimal representation (or representation

in base b)1 Since we expect intuitively that about one-tenth of the digits (in base

10) ...

over all integers, diverges. This suggests a way of describing the size of a set S of

integers. Let us say that S is "small" if 21/n, summed over S, converges, and

otherwise that S is "large." The

**set of integers**that contain no zeros at all is small.Can we say that a

**set of integers**must be small if each integer in the set containseither too few or too many zeros in its decimal representation (or representation

in base b)1 Since we expect intuitively that about one-tenth of the digits (in base

10) ...

Page 56

As we have just seen, questions like "How likely is it that a random

divisible by 57" sometimes have intuitive answers that can be made to seem

reasonable by calculating the densities of the

ago somebody thought of asking the question, "How likely is it that a positive

way and read from left to right.) The interesting thing about this question is that

most people will ...

As we have just seen, questions like "How likely is it that a random

**integer**isdivisible by 57" sometimes have intuitive answers that can be made to seem

reasonable by calculating the densities of the

**sets**in question. A number of yearsago somebody thought of asking the question, "How likely is it that a positive

**integer**has its first digit 1?" (The**integer**is supposed to be written in the usualway and read from left to right.) The interesting thing about this question is that

most people will ...

Page 58

Dividing by log 10"* = m log 10, we have UXOf") / mlogl0 almost equal to 1og2/

logl0=log102. It is relatively easy, but tedious, to take care of all the "almost" 's.

However, we shall outline an alternate approach [26] that avoids most of the

computation and shows that our set 5 has density log102 for any definition of

density with the following desirable properties; ordinary density, of course, lacks

one of these properties. Let E and F be disjoint

Dividing by log 10"* = m log 10, we have UXOf") / mlogl0 almost equal to 1og2/

logl0=log102. It is relatively easy, but tedious, to take care of all the "almost" 's.

However, we shall outline an alternate approach [26] that avoids most of the

computation and shows that our set 5 has density log102 for any definition of

density with the following desirable properties; ordinary density, of course, lacks

one of these properties. Let E and F be disjoint

**sets of integers**; let 2E mean the**set of integers**...### What people are saying - Write a review

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