Weighted universality of periodic zeta-function Текст научной статьи по специальности «Математика»

a po-group, and the following condition holds: if 0 < a £ R, then ab << a (ba << a) for all b £ R.

Theorem 1. If R is a right K-ring, then there is the convex right ideal Ia for each element a > 0 in R, and every element u £ Ia has a representation u = b — c, where 0 < b < ka and 0 < c < la for some integers k > 0 and l > 0.

A ring R =< R, +, •, <> is called a lattice K-ring if R is a right K-ring and a left K-ring, and the group < R, +, <> is a lattice-ordered group.

Suppose R and S are lattice K-rings and f is a homomorphism of the ring R to the ring S. f is said to be an l-homomorphism if f preserves the lattice operations.

Theorem 2. Suppose R =< R, +, •, <> is a lattice K-ring, I is a convex directed subgroup of the group G =< R, +, <>, and e is the natural homomorphism of the group G to the quotient-group G/I. Then there exists the lattice K-ring R/I, and e is an l-homomorphism of the lattice K-ring R to the lattice K-ring R/I.

References

1. Fuchs L. Partially Ordered Algebraic Systems. Moscow : Mir, 1965.

WEIGHTED UNIVERSALITY OF PERIODIC ZETA-FUNCTION M. Stoncelis (Vilnius, Lithuania) E-mail: stoncelis@su.lt

Let s = a + it be a complex variable and let a = {am : m £ N} be a periodic sequence of complex numbers with minimal period k £ N. The periodic zeta-function ((s; a) is defined, for a > 1, by the Dirichlet series

TO

am

z (s; a) = £ ms,

ms

m= 1

and, in view of the equality,

k

1m

Z(s; a) = ksS amZ(s,mm), a> 1

m=1

where Z(s,a), 0 < a ^ 1, denotes the Hurwitz zeta-function, has a meromorphic continuation to the whole complex plane with a simple pole at the point s = 1. If a\ + • • • + ak = 0, then the function Z(s,a) is entire one.

The function Z(s; a), as the majority of zeta-functions, for some sequences a is universal in the Voronin sense, i.e., a wide class of analytic functions can be approximated by shifts Z(s + ir; a). The first results in this direction were obtained by B. Bagchi, J. Steuding, and by J. Sander and J. Steuding. In [1], the universality of Z(s; a) with a multiplicative sequence a has been considered. However, the universality of the function Z(s; a) is a rather complicated problem. There exist examples of sequences a when Z(s; a) is not universal.

In the report, we present a weighted universality theorem for the function Z(s; a). Let w(t) be a positive function of bounded variation on [T0, to), T0 > > 0, such that the variation Va6w on [a, b] satisfies the inequality Va6w ^ ^ cw(a) with a certain constant c > 0 for any subinterval [a,b] G [To, to). Define

A = r : sup |Z(s + ir; a) - f (s)| < ^ .

I seK J

Then the following statement is valid.

Theorem 1. Suppose that the sequence a is multiplicative (amn = aman for all (m,n) = 1). Let K be a compact subset of the strip {s £ C : 1/2 < a < < 1} with connected complement, and let f (s) be a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every e > 0,

References

1. Laurincikas ASiauciunas D. Remarks on the universality of the periodic zeta-function // Math. Notes. 2006. Vol. 80, № 3-4.