# Moscow Mathematical Journal

Volume 18, Issue 2, April–June 2018 pp. 349–366.

Joint Value Distribution Theorems for the Riemann and Hurwitz Zeta-Functions

**Authors**:
Antanas Laurinčikas (1)

**Author institution:**(1) Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania

**Summary: **

In the paper, a class of functions φ(*t*) is introduced such
that a given pair of analytic functions is approximated simultaneously
by shifts ζ(*s*+*i*φ(*k*)), ζ(*s*+*i*φ(*k*), α), *k*∈ℕ, of the Riemann and
Hurwitz zeta-functions with parameter α for which the set {(log *p*: *p* is prime), (log(*m*+α): *m*∈ℕ_{0})} is linearly independent over ℚ. The
definition of this class includes an estimate for φ(*t*) and φ'(*t*) as well as
uniform distribution modulo 1 of the sequence {*a*φ(*k*): *k*∈ℚ}, *a*≠0.

2010 Math. Subj. Class. Primary: 11M06, 11M35.

**Keywords:**Hurwitz zeta-function, Riemann zeta-function, uniform distribution modulo 1, universality, weak convergence

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