If $\pi(x) > \operatorname{Li(x)},$ is $\vartheta(x) > x$? Are the two inequalities (solutions to both of which are known to exist but not known exactly) equivalent, similar, or mostly unrelated?

$\vartheta(x)$ is the Chebyshev Theta Function.

  • $\begingroup$ Similar - for sure. Mostly unrelated - I would guess not. Equivalent - for specific $x$ I would bet not, but I suspect the existence of solutions to one inequality might imply existence for the other (just a guess though). $\endgroup$ – Wojowu Jan 23 '16 at 19:39
  • $\begingroup$ @Wojowu Thanks for the quick response! Would it be expected that the two regions where each inequality holds have considerable overlap? $\endgroup$ – Carl Schildkraut Jan 23 '16 at 20:03
  • $\begingroup$ It's a reasonable thing to conjecture. One could probably even ask for quantative results regarding these, but I am completely not into the topic to give any kind of reference. $\endgroup$ – Wojowu Jan 23 '16 at 20:48
  • $\begingroup$ @Wojowu That makes sense. Do you have any idea as to how one would go about proving (or disproving) such a statement? I tried using the connections from $\vartheta$ to $\psi$ to $\psi_0$ to $\Pi_0$ to $\pi_0$ to $\pi$, but by the time one's gone through all that it just ends up ugly and useless. $\endgroup$ – Carl Schildkraut Jan 23 '16 at 21:28

One can address this problem using tools from comparative prime number theory ("prime number races"). Assuming RH, the (logarithmic) limiting distribution of the vector-valued error function $$ \bigg( \frac{\pi(t)-\mathop{\rm Li}(t)}{\sqrt t/\log t} , \frac{\theta(t)-t}{\sqrt t} \bigg) $$ is supported on the diagonal $y=x$ in $\Bbb R^2$; with further assumptions on the imaginary parts of the zeros of $\zeta(s)$, that distribution will not assign the origin any mass. Consequently, the two inequalities are either simultaneously true or simultaneously false for all $x>0$ except for a set of density $0$.

It's unthinkable that the real number where $\pi(x)$ crosses $\mathop{\rm Li}(x)$ will magically be the same real number where $\theta(x)$ crosses $x$; that's what would be required for the pair of inequalities to be perfectly correlated. Even if you restrict $x$ to be an integer, it's still extremely unlikely. For instance, any theoretical argument that linked the two inequalities should also link the inequalities $$ \pi(x;q,a) > \frac{\mathop{\rm Li}(x)}{\phi(q)} \quad\text{and}\quad \theta(x;q,a) > \frac{x}{\phi(q)} $$ concerning primes in arithmetic progressions; yet examples where one such inequality holds but the other doesn't are easily found by hand.

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