Introduction
Consider $\vartheta$ to be the Chebyshev function, that is, $\vartheta(x)$ denotes, for $x\in\mathbb N$, the sum $\sum_{p\le x,\ p\text{ prime}} \ln p$.
I am interested in asymptotics for $$\vartheta(x+y)-\vartheta(x)$$ as $x,y\to\infty$, where we assume that $y$ is smaller than $x$, but larger than some power of $x$, that is, $y\in[x^\alpha, x]$ for some $\alpha\in(0,1)$. (This latter relation between $x,y$ and $x\to\infty$ will be assumed implicitly during the rest of the post.)
More specifically, I would like to know about results which show that
\begin{equation}\label{eq:main}\tag{1}\vartheta(x+y)-\vartheta(y)=y+O\left(\frac{y}{(\ln x)^A}\right)\end{equation} for some $A>0$.
I found a result in the literature which establishes \eqref{eq:main} for $\alpha>\frac{7}{12}$ and $A=\frac 1{44}$ (see the "advanced estimates" section below), and would like to inquire about results which sharpen this result, either by lowering $\alpha$, or by increasing $A$.
Naive estimates
It is known [1] that $$\vartheta(x) = x + O\left(\frac x{(\ln x)^4}\right),$$ from which one can deduce \begin{equation}\vartheta(x+y)-\vartheta(y) = y + O\left(\frac{x}{(\ln x)^4}\right).\label{eq:insufficient}\tag{2}\end{equation}
However, this is not very satisfying, as I would like to achieve an error term of the form $$O\left(\frac{y}{(\ln x)^A}\right)$$ for some $A>0$. Since $y$ can be of order $x^\alpha$ for $\alpha\in(0,1)$, it follows that \eqref{eq:insufficient} is insufficient.
Naive estimates under Riemann hypothesis
Under the Riemann hypothesis, we have [2] $$\vartheta(x)=x+O\left(\sqrt x (\ln x)^2\right),$$ from which we get $$\vartheta(x+y)-\vartheta(x) = y + O\left(\sqrt x (\ln x)^2\right).$$ The result is satisfying, as we get \eqref{eq:main} for $\alpha>\frac 12$ and any $A>0$.
However, it is unsatisfying that we need to assume the Riemann hypothesis.
Advanced estimates
Consider the prime counting function $\pi$, so that $\pi(z)$ denotes the number of prime numbers $p$ satisfying $p\le z$ for $z\in\mathbb R$.
We then have $$\vartheta(x+y)-\vartheta(x) = \left(\ln x+O\left(\frac yx\right)\right) (\pi(x+y)-\pi(x)).$$
Heath-Brown [3] has obtained $$\pi(x)-\pi(x-y) = \frac{y}{\ln x} + O\left(y (\ln x)^{-\frac{45}{44}}\right)$$ whenever $x^{\frac 7{12}}\le y\le x$.
Therefore, by replacing $x$ with $x+y$, this establishes \eqref{eq:main} for $\alpha>\frac{7}{12}$ and $A=\frac 1{44}$.
Advanced estimates which I didn't get to work
Heath-Brown [4] has established $$\pi(x)-\pi(x-y)=\frac{y}{\ln x}\left(1+O\left(\left(\frac{\ln \ln x}{\ln x}\right)^4\right)\right)$$ for $x^{\frac 7{12}}\le y\le\frac{x}{(\ln x)^4}$, which, by replacing $x$ with $x+y$, translastes straightforwardly to an analogous statement for $\pi(x+y)-\pi(x)$.
However, I don't know what happens when $\frac{x}{(\ln x)^4}<y\le x$...
References
[1] Pierre Dusart, Sharper bounds for $\psi, \theta,\pi, p_k$. Rapport de recherche n° 1998-06.
[2] Helge von Koch, Sur la distribution des nombres premiers, Acta Math. v. 24, pages. 159-182, 1901.
[3] David Rodney Heath-Brown, Sieve identities and gaps between primes. Astérisque, tome 94, pages 61-65, 1982.
[4] David Rodney Heath-Brown, The number of primes in a short interval. Journal für die reine und angewandte Mathematik 389, pages 22-63, 1998.
