Multiplicative functions on the set of shifted prime numbers

This paper is concerned with the behavior of multiplicative functions on the set $\{p+a\}$, where $p$ is a prime and $a$ is a nonzero integer. Several results are obtained in which either the average value of a multiplicative function on this set is estimated or its asymptotic behavior is determined. As one application a nontrivial estimate of
$$\sum\limits_{p\leqslant x}\chi_q(p+a)$$
is found, where $x\geqslant q^{1/4+\varepsilon}$, $q$ is a sufficiently large prime, and $\chi$ is a character of degree greater than 4.