A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions

For j=1,…,r, let Qj be a positive definite nj×nj matrix, and ζ(sj;Qj) denote the corresponding Epstein zeta-function. In this paper, assuming that nj⩾4 is even and x̲TQjx̲∈Z,x̲∈Zr∖{0̲}, a joint limit theorem of Bohr–Jessen type for the functions ζ(s1;Q1),…,ζ(sr;Qr), by using generalizing shifts ζ(σ1+iφ1(t);Q1),…,ζ(σr+iφr(t);Qr), is proved. Here, the functions φ1(t),…,φr(t) are increasing to +∞, with monotonic derivatives φj′(t) satisfying the asymptotic growth conditions: φj(t)≪tφj′(t), and φj′(t)=o(φj+1′(t)) as t→∞. An explicit form of the limit measure is given. This theorem extends and generalizes the previous result on the joint value-distribution of Epstein zeta-functions.