Multiplicative functions with bounded seminorm on the set of shifted primes (Q2743912)

For a subset \(A\subseteq\mathbb{N}\) and real numbers \(\alpha\geq 1\) let \(L_\alpha(A)\) be the space of functions \(f\) with bounded seminorm NEWLINE\[NEWLINE\|f\|_{\alpha,A}= \Biggl\{\limsup_{x\to\infty} {1\over A(x)}\sum_{\substack{ n\leq x\\n\in A}}\bigl|f(n)\bigr|^\alpha \Biggr\}^{1/\alpha}NEWLINE\]NEWLINE on \(A\) and \(L^*(A)\) the set of functions which are uniformly summable on \(A\). In the present paper the authors investigate connections between the classes \(L_\alpha (\mathbb{N})\), \(L^*(\mathbb{N})\), \(L_\alpha (\mathbb{P}+1)\), \(L^*(\mathbb{P}+1)\) where \(\mathbb{P}+1= \{p+1:p\) prime\} denotes the set of shifted primes.NEWLINENEWLINENEWLINEThe conjecture that a multiplicative function \(f\in L_\alpha\), \(\alpha>1\), belongs to \(L_1(\mathbb{P}+1)\) is proved under some additional conditions.