Narrow Results

In this paper, the mean values of the recurrence are computed for general group actions. Let $(X,d)$ be a metric space with a finite measure $\mu$ and $G$ be a countable group acting on $(X,d)$. Let $F_1,F_2,\dots$ be a sequence of subsets of $G$ with $\left|F_n\right|\to \infty$ and put $E_n=F_n^{-1}{F}_n$. If the Hausdorff measure $H_h$ is finite on $X$ and $\mu$ is $T$-invariant. We assume that $\mu$ and $H_h$ are concordant. Then the function

In this paper, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and a generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.

Let $k$ be a fixed integer, we define the multiplicative function $D_k(n)=\frac{d_k(n)}{d_k^{\ast }(n)}$, where $d_k(n)$ is the Piltz divisor function and $d_k^{\ast }$ is the unitary analogue function of $d_k$. The main purpose of this paper to use elementary methods to study the mean value of the function $D_k(n)$.