TITLE: $π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?
QUESTION [9 upvotes]: (I posted this question on Math SE but it has had no answer for a year now so I would like to ask if anyone here can provide one.)
Thinking about the prime number theorem, I wondered whether it is known that there is some constant $c$ such that $π(x+y) ≤ π(x) + c·y/\ln(y)$ for every integers $x,y > 1$. I read that experts believe $π(x+y) ≤ π(x) + π(y)$ fails for some $y$, since it fails for $y = 3159$ if the k-tuple conjecture holds, but it is just barely false, so I am curious if it is known to be true if the inequality is relaxed by a constant factor. If so, is it also known that $π(x+y) ≤ π(x) + π(y) + c·\!\sqrt{y}·\ln(y)$ for some constant $c$? I simply do not know how to search for such conjectures, and neither Wikipedia nor Wolfram seem to state any results that would affirm or refute these two conjectures easily, so any references would be appreciated!

REPLY [15 votes]: As mentioned in my comment, Montgomery and Vaughan (The Large Sieve, Mathematika 20 (1973) 119–132, doi:10.1112/S0025579300004708) showed an explicit version of the Brun--Titchmarsh inequality:
$$ 
\pi(x+y) - \pi(x) \le \frac{2y}{\log y}.
$$
The other question asked is likely false.  Hensley and Richards (Primes in intervals, Acta Arithmetica 25 (1974) 375-391, EuDML) showed that the Hardy-Littlewood $k$-tuples conjecture contradicts the hypothesis that $\pi(x+y) -\pi(x) \le \pi(y)$.  In fact their paper establishes that (on the $k$-tuples conjecture) one has (for large fixed $y$, and infinitely many $x$)
$$ 
\pi(x+y) -\pi(x) \ge \pi(y) + (\log 2 -\epsilon) \frac{y}{(\log y)^2}. 
$$
Roughly the lower bound that they are obtaining is $2 \pi(y/2)$ (and some conjectural improvements over this are also discussed).