what is the Best known Upper bound on Twin Primes?

Are there proven results for a lower and upper bound of the modular prime counting function?

• For example, is there a proven lower and upper bound for the number of primes less than x that are congruent to 1(mod3)? In general, for numbers a and b, is there a proven lower and upper bound for the number of primes less than x that are congruent to a(modb)? I conjectured : 1) if gcd(a,b)=1, then the number of primes less than x that are congruent to a(modb) approaches PI(x)/φ(b) as x approaches infinity. 2) if gcd(a,b) is not 1 and a is prime, then the number of primes less than x that are congruent to a(modb)=1. 3) otherwise the number of primes less than x that are congruent to a(modb) is 0 unless b is prime and a is 0 in which case it is 1. Has something like this conjecture been proven? Still, this conjecture does not give a lower and upper bound though it may be close to finding one. Even so, proving a lower and upper bound seems much more difficult, and I was wondering if someone has already done this. In the places where I said, the number of primes less than x, I should have said the number of primes less than or equal to x. I think that's what the modular prime counting function is. 2) requires x =a. Otherwise, we get 0, not 1. The = sign should be replaced with the word "is." 3) when b is prime and a is 0 we need x =b. Otherwise, we get 0.

• Answer:

  1) http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions The theorem states exactly that.