There are several results on this topic in the article [S. B. Stechkin, A. Yu. Popov, “The asymptotic distribution of prime numbers on the average”, Uspekhi Mat. Nauk, 51:6 (312), 21-88 (1996)][1]. An English translation is [here][2].


  [1]: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=1018&option_lang=eng
  [2]: https://iopscience.iop.org/article/10.1070/RM1996v051n06ABEH003000/pdf