# What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$

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?

Thanks,

-Larry

• 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 – joro Apr 23 '13 at 13:40
• Doesn't Nagura give $\psi(x) > 0.916x - 2.318$ ? – lhf Nov 12 '15 at 0:13
• @lhf, Yes, you are right. – Larry Freeman Nov 12 '15 at 0:29

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.