**Find all values of such that there exists a function that satisfies the following properties:**

**1), if and only if .**

**2), .**

Answer: .

Proof: does not fit into the criteria because it requires . Fix .

**Bertrand’s Postulate (a.k.a. Tchebyshev’s theorem):** , with prime.

Now we aim to find a such that .

For , take . For odd, , apply Tchebyshev’s theorem to , there exists a prime . But now is even and thus is not prime, and thus we have , which shows that is the desired prime number as . For even, apply Tchebyshev’s theorem to to yield a prime which gives . The has been chosen for all .

Let , . Let .

(E.g. let , then .

Define an equivalence relation on : for , if and only if . It is easy to verify that this is indeed an equivalence relation. Partition into equivalence classes , by this relation. Let be indexed by the set .

**Axiom of Choice:** If is a family of sets indexed by the set , then the Cartesian product is non-empty.

Apply Axiom of Choice to the equivalence classes of , and we get a family of elements , with each corresponding to exactly one .

Let be the -adic evaluation of . (That is, reduce to simplest form , if does not divide , and if .

Let . For , suppose . Define . (E.g. if .Similarly, . This gives .) We set

Indeed, we show first show that the function satisfies the conditions.

First, we see that does not vanish for non-zero .

We claim that if . This is obvious by substituting the definitions of and , and then LHSRHS.

Now, we verify the conditions. Condition 1 is already checked. If , . If , let . For with , we have , and thus . Since , there are such s. .

Thus . We have found the desired function, for each of .

Q.E.D.

**Comment**: to be written later.

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