Remarks on the universality of the periodic zeta function

A Weighted Universality Theorem for Periodic Zeta-Functions

“…and, for k = 0, using properties of w(t), we find that Since the series for ζ n (s, ω; a) is absolutely convergent for σ > 1 2 [11], the function u n is continuous one. We setP n = m H u −1 n , where, for A ∈ B(H(D)), Lemma 2.…”

“…We note that the assumptions of Theorem 1 imply that the sequence a is not multiplicative. We recall that the sequence a is multiplicative if a mn = a m a n for all coprime m, n ∈ N. The universality of ζ(s; a) with multiplicative sequence a was obtained in [11]. Denote by H 0 (K), K ∈ K, the class of continuous non-vanishing functions on K, which are analytic in the interior of K. Theorem 2.…”

“…Denote by H 0 (K), K ∈ K, the class of continuous non-vanishing functions on K, which are analytic in the interior of K. Theorem 2. [11]. Suppose that the sequence is multiplicative and for all primes p. Let K ∈ K and f (s) ∈ H 0 (K).…”

A SURVEY ON THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

“…Later, Laurinčikas also obtained universality theorems for Matsumoto zeta-functions 1 ( [78] under a strong assumption), and for the zetafunction attached to Abelian groups ([82] [83]). Laurinčikas andŠiaučiūnas [122] proved the universality for the periodic zeta-function ζ(s, A) = ∞ n=1 a n n −s (where A = {a n } ∞ n=1 is a multiplicative periodic sequence of complex numbers) when the technical condition ∞ m=1 |a p m |p −m/2 ≤ c (3.1) (with a certain constant c < 1) holds for any prime p. Schwarz, R. Steuding and J. Steuding [192] proved another universality theorem on certain general Euler products with conditions on the asymptotic behaviour of coefficients.…”

Совместная Универсальность Дзета-Функций С Периодическими Коэффициентами

“…Универсальность функции ζ(s; a) в случае, когда последовательность a муль-типликативна (a 1 = 1 и a mn = a m a n для всех m, n ∈ N, (m, n) = 1), была получена в [15].…”