Let me note that, in the past few days, we seem to have found a negative answer Question 2 (in Section 6) for the part concerning the “shift-invariant case”, but the construction is fairly long and complicated, so we have thought to write a second paper on this. On the other hand, we do not yet know whether the other part of the same question (i.e., “Does there exist a monotone, subadditive, and (-1)-homogeneous function $f: P({\bf N}) \to \bf R$ such that $f(\emptyset) = 0$, $f({\bf N}) = 1$, and $f$ has the weak, but not the strong, Darboux property?”) has a negative answer too. If you have any idea, comment, or whatever on this or other questions, please just let me know.have thought to write a second paper on this. On the other hand, we do not yet know whether the other part of the same question (i.e., “Does there exist a monotone, subadditive, and (-1)-homogeneous function $f: P({\bf N}) \to \bf R$ such that $f(\emptyset) = 0$, $f({\bf N}) = 1$, and $f$ has the weak, but not the strong, Darboux property?”) has a negative answer too. If you have any idea, comment, or whatever on this or other questions, please just let me know.

]]>For the record, I would be very happy with an answer (either positive or not) to Questions 7 and 8, in spite of the fact that they are admittedly much less interesting than others. But a counterexample to either of them, if it exists, is probably rather complicated, if you think of the construction that was necessary to prove the existence of a non-monotone upper quasi-density (Theorem 1).

]]>where is a real number and is the Riemann zeta function. These (probability) measures have the lovely property that being divisible by and being divisible by , for two distinct primes , are independent events. Moreover, as they converge to a related measure called the logarithmic density, when it exists. The logarithmic density itself has the further desirable property that if it and the natural density both exist then they must agree.

This can be used to explain why, for example, the probability (in a suitable sense) that two random integers are relatively prime is . See, for example, this blog post.

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