In 1934 Romanov showed that a positive proportion of the natural numbers can be written as the sum of a prime and a power
of two. Yong-Gao Chen and Xue-Gong Sun proved recently that the lower asymptotic density of this set is larger than 0.0868.
We improve this bound to 0.09368 and show various connections with the generalized twin prime problem and the Goldbach-Linnik
In the 1980’s the author proved lower bounds for the mean value of the modulus of the error term of the prime number theorem and other important number theoretic functions whose oscillation is in connection with the zeros of the Riemann zeta function. In the present work a general theorem is shown in a simple way which gives a lower bound for the mentioned mean value as a function of a hypothetical pole of the Mellin transform of the function. The conditions are amply satisfied for the Riemann zeta function. In such a way the results recover the earlier ones (even in a slightly sharper form). The obtained estimates are often optimal apart from a constant factor, at least under reasonable conditions as the Riemann Hypothesis. This is the case, in particular, for the error term of the prime number theorem.