I just found the following sentence from the #P wiki page:
"Jerrum, Valiant, and Vazirani showed that every #P-complete problem either has an FPRAS, or is essentially impossible to approximate; if there is any polynomial-time algorithm which consistently produces an approximation of a #P-complete problem which is within a polynomial ratio in the size of the input of the exact answer, then that algorithm can be used to construct an FPRAS."
the referece  is Mark R. Jerrum; Leslie G. Valiant; Vijay V. Vazirani (1986). "Random Generation of Combinatorial Structures from a Uniform Distribution". Theoretical Computer Science (Elsevier) 32: 169–188.
I took a quick look at . But it seems to me that the results of  do not contain anything similar to what is in the wiki page. Is there a mistake in the wiki page? Thanks.