Riemann Hypothesis Conjectures

Our program made the following conjectures about |\pi(x)-Li(x)|, the difference in absolute value of the number of primes no more than x and the logarithmic integral bounded below by 2. It is a result of Koch (1901) that the Riemann Hypothesis is equivalent to the statement that |\pi(x)-Li(x)|\leq \sqrt{x}\ln x for x>2.01. So we thought it could be interesting to generate conjectures for this quantity. We got several. Here are 3 that we tested further.

  1. |\pi(x)-Li(x)| \leq -digits10(x)\mathbin{\char`\^}(1/4) + digits2(x)
  2. |\pi(x)-Li(x)| \leq sqrt(x)-\log(euler\_phi(x))
  3. |\pi(x)-Li(x)|\leq maximum(digits10(x), 1/2*sqrt(previous\_prime(x)))

Here digits10(x) is the number of digits in the base-10 representation of x, euler_phi(x) is the value of euler’s phi function, and previous_prime(x) is the largest prime lexx than x. These statements were tested for all integers x from 3 to 1,000,000. Of course, this doesn’t prove these conjectures. But they might imply the Riemann Hypothesis.

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