Number of integers in an assigned A. P., ≤ and prime to primes greater than ^{}

@inproceedings{Ramaswami1951NumberOI,
  title={Number of integers in an assigned A. P., ≤ and prime to primes greater than ^\{\}},
  author={V. Ramaswami},
  year={1951}
}
In the following: x denotes any real number greater than 1; y, c, t denote real positive numbers; I= log (x); d, n, m, v, k denote integers satisfying n>0, d>0, m>0, 0 xc, and a problem of S. S. Pillai; 2r(v, k, x) denotes the number of primes less than or equal to x in A(1, v, k); P(v, k, x) -p_x,p_(modk) p-l, p denoting a prime number; n(v, k) denotes the least positive integer satisfying nn(v,k) v mod k for (n, k)=1. [We note that I F(x, k)j < I/(k, k).] Buchstab2 has proved the result