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Meromorphic continuation of Dirichlet series connected with the distribution of numbers modulo 1. (English. Russian original) Zbl 0943.30002
Lupanov, O. B. (ed.), Analytic number theory and applications. Collected papers in honor of the sixtieth birthday of Professor Anatolii Alexeevich Karatsuba. Moscow: MAIK Nauka/Interperiodica Publishing, Proc. Steklov Inst. Math. 218, 340-351 (1997); translation from Tr. Mat. Inst. Steklova 218, 343-353 (1997).
Let \(M\) denote the set of all functions which are analytic in the halfplane \(\text{Re }s>1\) and which can be continued meromorphically to the whole \(s\)-plane. Then, using very classical analytic tools, the author proves that for any integer \(q \geq 2\) and for an arbitrary polynomial \(P\) in one variable both Dirichlet series \(\sum^\infty_{n=1} P(\{n^{1/q}\})n^{-s}\) and \(\sum^\infty_{n=1} P(\{\log_q n\})n^{-s}\) belong to \(M\). For the special case \(P(x) = x\) the location of the poles of the corresponding function is completely determined in both situations.
Another theorem says that, if \(q\geq 2\) is an integer, there exists an infinite set of real irrational numbers \(\alpha\) for which \(\sum^\infty_{n=1}\{\alpha q^n\}n^{-s}\in M\). Again, for each such \(\alpha\) the location of the poles of the corresponding function can be given explicitly. Of course, \(\{t\}\) denotes the fractional part \(t-[t]\) of a real number \(t\).
For the entire collection see [Zbl 0907.00013].
30B40 Analytic continuation of functions of one complex variable
11J71 Distribution modulo one
11M41 Other Dirichlet series and zeta functions
30B50 Dirichlet series, exponential series and other series in one complex variable