Primes between $x$ and $x+x^\theta$

Question

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
• Prime powers between $x$ and $x+x^\theta$ 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. Journal of the Mathematical Society of Japan, 34:1, pp. 95–123, 1982.

Answer 1

I found one paper that improves on the quoted result (in a narrow range): $$\eta(\theta )=\frac{100\theta -45}{11}\qquad\text{is admissible for}\qquad \frac{6}{11}<\theta \le \frac{11}{20}.$$ See Lou-Qi: Upper bounds for primes in intervals (Chinese), Chinese Ann. Math. Ser. A 10 (1989), 255-262.

Answer 2

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, 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.