Iwaniec [1] proved that
$$
\pi(x+x^\theta)-\pi(x) < \frac{(2+\varepsilon)x^\theta}{\eta(\theta)\log x},\ x>x\_0(\varepsilon,\theta).
$$
with
$$
\eta(\theta)=\frac{15\theta-2}{9}.
$$


(Actually, he proves that a function $\eta(\theta)>\theta$ exists, and that this is an admissible choice. This choice gives nontrivial information for $\theta>1/3.$ He gives others like $\eta\_1(\theta)=(1+\theta)/2$ for $\theta>1/2.$)


Two other questions have asked about the primes in this interval


* [Prime Power Gaps](https://mathoverflow.net/q/111029/6043)
* [Prime powers between $x$ and $x+x^\theta$](https://mathoverflow.net/q/262162/6043)
but neither asks about upper bounds, nor do answers give information.


Fundamentally, I'd like information on
$$
f(\theta) := \limsup\_{x\to\infty} \frac{\pi(x+x^\theta)-\pi(x)}{x^\theta/\log x}
$$


What modern results are available? The result above is $f(\theta) \le 18/(15\theta-2)$ for $1/3<\theta<1.$


[1] Henryk Iwaniec. [On the Brun–Titchmarsh theorem](https://doi.org/10.2969/jmsj/03410095). *Journal of the Mathematical Society of Japan*, **34**:1, pp. 95–123, 1982.



Primes between $x$ and $x+x^\theta$
Montgomery (1) gives a list of 40 exponent pairs $(\kappa,\lambda)$ which can be plugged into Iwaniec's formula


$$
\eta(\theta)=\left(1+\frac{1-\lambda+2\kappa}{3-\lambda-\kappa/2}\right)\theta - \frac{\kappa}{3-\lambda-\kappa/2}
$$


to yield bounds for $0<\theta\le1/2.$ Of these, 36 are optimal in some interval; adding the zeta function value for $\theta>1/2$ yields


$$
\eta(\theta)=\begin{cases}
\theta,&\text{ if }0<\theta\le1/9\\
\frac{1047\theta-2}{1029},&\text{ if }1/9\le\theta\le514/3597\\
\frac{531\theta-2}{515},&\text{ if }514/3597\le\theta\le322/2061\\
\frac{369\theta-2}{354},&\text{ if }322/2061\le\theta\le194/1101\\
\frac{444\theta-4}{417},&\text{ if }194/1101\le\theta\le130/669\\
\frac{189\theta-2}{176},&\text{ if }130/669\le\theta\le46/219\\
\frac{498\theta-8}{451},&\text{ if }46/219\le\theta\le2/9\\
\frac{228\theta-4}{205},&\text{ if }2/9\le\theta\le362/1569\\
\frac{1161\theta-22}{1037},&\text{ if }362/1569\le\theta\le182/743\\
\frac{1167\theta-26}{1027},&\text{ if }182/743\le\theta\le52/201\\
\frac{921\theta-22}{805},&\text{ if }52/201\le\theta\le342/1291\\
\frac{254\theta-8}{215},&\text{ if }342/1291\le\theta\le226/833\\
\frac{669\theta-22}{563},&\text{ if }226/833\le\theta\le118/415\\
\frac{116\theta-4}{97},&\text{ if }118/415\le\theta\le186/641\\
\frac{589\theta-22}{487},&\text{ if }186/641\le\theta\le94/303\\
\frac{587\theta-26}{473},&\text{ if }94/303\le\theta\le310/959\\
\frac{1439\theta-66}{1153},&\text{ if }310/959\le\theta\le286/855\\
\frac{1327\theta-66}{1049},&\text{ if }286/855\le\theta\le127/376\\
\frac{351\theta-22}{265},&\text{ if }127/376\le\theta\le74/217\\
\frac{393\theta-26}{293},&\text{ if }74/217\le\theta\le45/128\\
\frac{325\theta-22}{241},&\text{ if }45/128\le\theta\le62/171\\
\frac{347\theta-26}{251},&\text{ if }62/171\le\theta\le37/98\\
\frac{281\theta-22}{201},&\text{ if }37/98\le\theta\le66/173\\
\frac{823\theta-66}{585},&\text{ if }66/173\le\theta\le426/1093\\
\frac{317\theta-26}{224},&\text{ if }426/1093\le\theta\le682/1733\\
\frac{1243\theta-114}{851},&\text{ if }682/1733\le\theta\le438/1103\\
\frac{1601\theta-150}{1089},&\text{ if }438/1103\le\theta\le36/89\\
\frac{1651\theta-162}{1107},&\text{ if }36/89\le\theta\le486/1181\\
\frac{659\theta-66}{439},&\text{ if }486/1181\le\theta\le474/1141\\
\frac{1465\theta-150}{969},&\text{ if }474/1141\le\theta\le1626/3865\\
\frac{595\theta-66}{383},&\text{ if }1626/3865\le\theta\le120/281\\
\frac{1427\theta-162}{911},&\text{ if }120/281\le\theta\le422/973\\
\frac{1249\theta-150}{781},&\text{ if }422/973\le\theta\le846/1921\\
\frac{923\theta-114}{571},&\text{ if }846/1921\le\theta\le1542/3469\\
\frac{470\theta-60}{287},&\text{ if }1542/3469\le\theta\le34/75\\
\frac{15\theta-2}{9},&\text{ if }34/75\le\theta\le1/2\\
\frac{\theta+1}{2},&\text{ if }1/2<\theta\le7/12\\
2,&\text{ if }7/12<\theta<1.
\end{cases}
$$


This is hardly a satisfactory answer, giving only 20-year-old results and neglecting Vinogradov's method for small $\theta,$ but it may be useful so I'll leave it here.


I’ve added Huxley’s result
$$
\pi(x)-\pi(x-y) \sim \frac{y}{\log x}\text{ for }x^\theta \le y \le x/2
$$
with $\theta>7/12$.


1. Hugh L. Montgomery, [Harmonic Analysis as found in Analytic Number Theory](https://www.nato-us.org/analysis2000/papers/montgomery.pdf), in Twentieth Century Harmonic Analysis—A Celebration, Springer, 2001
2. M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), pp. 164-170.


