I was reading through Jitsuro Nagura's [proof](https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-28/issue-4/On-the-interval-containing-at-least-one-prime-number/10.3792/pja/1195570997.full) 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](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-6/issue-1/Approximate-formulas-for-some-functions-of-prime-numbers/10.1215/ijm/1255631807.full) 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



What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$
The most recent results on bounds for $\psi(x)$ are from this year:


[Sharper estimates for Chebyshev's functions $\vartheta$ and $ψ$](http://arxiv.org/abs/1302.7208), 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.
> 
> 
> 


