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Cosmological observations, e.g., cosmic microwave background, have preciselymeasured the spectrum of primordial curvature perturbation on larger scales,but smaller scales are still poorly constrained. Since primordial black holes(PBHs) could form in the very early Universe through the gravitational collapseof primordial density perturbations, constrains on the PBH could encodes muchinformation on primordial fluctuations. In this work, we first derive a simpleformula for lensing effect to apply PBH constraints with the monochromatic massdistribution to an extended mass distribution. Then, we investigate the latestfast radio burst observations with this relationship to constrain two kinds ofprimordial curvature perturbation models on the small scales. It suggests that,from the null search result of lensed fast radio burst in currently availableobservations, the amplitude of primordial curvature perturbation should be lessthan $8\times 10^{-2}$ at the scale region of $10^5-10^6~\rm Mpc^{-1}$. Thiscorresponds to an interesting mass range relating to binary black holesdetected by LIGO-Virgo-KAGRA and future Einstein Telescope or Cosmic Explorer.

Exploring primordial curvature perturbation on small scales with the lensing effect of fast radio bursts