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An inductive proof is typically done with three steps:

  • Base Case
  • Inductive Hypothesis
  • Inductive Step

The argument that follows does not have a Base Case.

  • Is this approach valid (I believe that it is -- but I could be wrong)
  • Assuming this approach is valid, is my use of this approach valid (there could be a mistake in my reasoning)

Let:

$$H_n(x) = \sum\limits_{i |n\#}\left\lfloor\frac{x}{i}\right\rfloor\mu(i)$$

where:

  • $H_n(x)$ is the count of $t$ where $t \le x$ and gcd$(t,n\#)=1$
  • $n\#$ is the primorial of $n$
  • $\mu(i)$ is the möbius function.

If $p_k$ is the $k$th prime, then:

$$H_{p_{k+1}}(x) = H_{p_k}(x) - H_{p_k}\left(\frac{x}{p_{k+1}}\right)$$

I was wondering if it is valid to use the following inductive argument with recurrent relations $W_k(n), d_k(n), c_k(n)$ where I do not define $W_0(n), c_0(n), d_0(n)$.

Let:

  • $W_k(x) = W_{k-1}(x) - \dfrac{W_{k-1}(x)}{p_k}$
  • $c_k = c_{k-1} + \dfrac{d_{k-1}}{p_k}$
  • $d_k = d_{k-1} + \dfrac{c_{k-1}}{p_k}$
  • $U_k(x) = W_k(x) + c_k$
  • $L_k(x) = W_k(x) - d_k$

Rather than defining $W_0(n), c_0, d_0$ which may or may not exist, assume that the following is valid up to $k$:

  • $U_k(x) \ge H_{p_k}(x) \ge L_k(x)$

  • $\dfrac{U_k(x)}{w} \ge H_{p_k}\left(\dfrac{x}{w}\right) \ge \dfrac{L_k(x)}{w}$ for all $1 \le w \le x$

Then:

  • $H_{p_{k+1}}(x) \le U_k(x) - \dfrac{L_k(x)}{p_{k+1}} \le W_k(x) - \dfrac{W_k(x)}{p_{k+1}} + c_k + \dfrac{d_k}{p_{k+1}} = W_{k+1}(x) + c_{k+1} = U_{k+1}(x)$
  • $H_{p_{k+1}}(x) \ge L_k(x) - \dfrac{U_k(x)}{p_{k+1}} \ge W_k(x) - \dfrac{W_k(x)}{p_{k+1}} - d_k - \dfrac{c_k}{p_{k+1}} = W_{k+1}(x) - d_{k+1} = L_{k+1}(x)$
  • $H_{p_{k+1}}\left(\dfrac{x}{w}\right) \le \dfrac{U_k(x) - \frac{L_k(x)}{p_{k+1}}}{w} \le \dfrac{W_k(x) - \frac{W_k(x)}{p_{k+1}} + c_k + \frac{d_k}{p_{k+1}}}{w} = \dfrac{W_{k+1}(x) + c_{k+1}}{w} = \dfrac{U_{k+1}(x)}{w}$
  • $H_{p_{k+1}}\left(\dfrac{x}{w}\right) \ge \dfrac{L_k(x) - \frac{U_k(x)}{p_{k+1}}}{w} \ge \dfrac{W_k(x) - \frac{W_k(x)}{p_{k+1}} - d_k - \frac{c_k}{p_{k+1}}}{w} = \dfrac{W_{k+1}(x) - d_{k+1}}{w} = \dfrac{L_{k+1}(x)}{w}$

Does this argument hold even if existence has not been proven?


Edit 1:

Adding details based on a flag that it is unclear what I am asking. Hopefully, this helps (details as a blockquote).

Please add a comment if the additional details do not help.


Edit 2:

Shortened the question based on a comment to my question here.

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Yes, your argument does hold without proving a base case. Though you should make very clear that your argument is really just the inductive step;

If there exist $W_k(x)$, $c_k$, $d_k$, $U_k(x)$ and $L_k(x)$ satisfying the given recurrence relations, and

if the given inequalities hold for $k$, then they hold for $k+1$.

This does not prove that any of the inequalities hold; it does not even prove that such sequences exist. It does now suffice to prove a base case, for example by exhibiting such $W_0(x)$, $c_0$, $d_0$, $U_0(x)$ and $L_0(x)$.

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