(I posted [this question on Math SE](https://math.stackexchange.com/q/3327182/21820) 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](https://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples), 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](https://en.wikipedia.org/wiki/Prime_number_theorem) nor [Wolfram](http://mathworld.wolfram.com/PrimeNumberTheorem.html) seem to state any results that would affirm or refute these two conjectures easily, so any references would be appreciated!



$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?
As mentioned in my comment, [Montgomery and Vaughan](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/152543/mtks0025579300004708.pdf?sequence=1&isAllowed=y) (*The Large Sieve*, Mathematika **20** (1973) 119–132, doi:[10.1112/S0025579300004708](https://doi.org/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](http://matwbn.icm.edu.pl/ksiazki/aa/aa25/aa2548.pdf) (*Primes in intervals*, Acta Arithmetica **25** (1974) 375-391, [EuDML](https://eudml.org/doc/205282)) 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).


