A mean-value theorem for multiplicative functions on the set of shifted primes
The main theorem of the authors is: Let \( f: {\mathbb N} \to {\mathbb C} \) be a multiplicative function with bounded 2-norm \[ \|f \|_2 = \limsup_{x \to \infty} \biggl( { 1 \over x} \sum_{n \leq x} |(f(n)|^2 \biggr)^{1 \over 2} < \infty. \] Assume that there is a Dirichlet character \( \chi \bmod d \), for which the mean-value \( M(f \cdot \chi) \) exists and is \( \not= 0 \), and assume that the values \( |f(p)|\) are not too big, \[ \sum_{p \leq x} |f(p)|^2 \leq A \cdot x \cdot (\log x)^{-\rho} \quad \text{for some } \rho > 0. \] Then \[ m_p(f) = \lim_{x \to \infty} {1 \over \pi(x)} \cdot \sum_{p \leq x} f(p+1) \] exists, and a product representation for \( m_p(f) \) is given. Moreover, if \( M(|f|) \) exists, then \( m_p(|f|) \) exists, too. The main lemma is an extension of a result of the second author [\textit{N. M. Timofeev}, Math. USSR, Izv. 39, 1189-1207 (1992); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, 1238-1256 (1991; Zbl 0778.11052)]. It gives an asymptotic formula for \[ {1 \over \pi(x)} \cdot \sum_{p \leq x} f(p+1) \] for functions \( f \) satisfying the assumptions of the theorem, and satisfying in addition \[ \sum_{p \leq x} \Bigl|1 - \chi^*(p) f(p) \Bigr|\cdot {\log p \over p} = o (\log x) \] for some primitive character \( \chi^* \).
mean-values of complex-valued multiplicative functions
mean-values on the set of additively shifted primes
application of the theorem of Daboussi-Elliott-Indlekofer
scientific article; zbMATH DE number 1203260