Joint weighted universality of the Hurwitz zeta-functions

Joint weighted universality theorems are proved concerning simultaneous approximation of a collection of analytic functions by a collection of shifts of Hurwitz zeta-functions with parameters α 1 , … , α r \alpha _1,\dots ,\alpha _r . For this, linear independence is required over the field of rational numbers for the set { log ⁡ ( m + α j ) : m ∈ N 0 = N ∪ { 0 } , j = 1 , … , r } \{\log (m+\alpha _j)\,:\, m\in \mathbb {N}_0=\mathbb {N}\cup \{0\},\;j=1,\dots ,r\} .