I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$.

In the paper, he uses the following bounds for the second Chebyshev function $\psi(x)$:

$$1.086x > \psi(x) > 0.916x - 6.954$$

If I apply the better upper bound from Rosser & Schoenfeld, 1962 of:

$$1.03883x > \psi(x)$$

Then Nagura's proof shows that there is always a prime between $x$ and $\frac{8x}{7}$ when $x \ge 34$.

Is this the best upper and lower bound for $\psi(x)$:

$$1.03883x > \psi(x) > 0.916x - 6.954$$

Does anyone know of any results that improve on these bounds?



  • $\begingroup$ Under RH there is better bound for $|x - \psi(x)|$. ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0457374-X Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$. II $\endgroup$
    – joro
    Apr 23, 2013 at 13:40
  • 2
    $\begingroup$ Doesn't Nagura give $\psi(x) > 0.916x - 2.318$ ? $\endgroup$
    – lhf
    Nov 12, 2015 at 0:13
  • $\begingroup$ @lhf, Yes, you are right. $\endgroup$ Nov 12, 2015 at 0:29

1 Answer 1


The most recent results on bounds for $\psi(x)$ are from this year:

Sharper estimates for Chebyshev's functions $\vartheta$ and $ψ$, February 2013.

In this article we present some improved results for Chebyshev's functions $\vartheta$ and $\psi$ using the new zero-free region obtained by H. Kadiri and the first $10^{13}$ zeros of the Riemann zeta function on the critical line calculated by Xavier Gourdon. The methods in the proofs are similar to those of the Rosser-Shoenfeld papers on this subject.


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