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Probabilistic renormalization and analytic continuation. (English) Zbl 1511.11076

In the paper under review, the authors introduce a theory of probabilistic renormalization for series. The renormalized values are encoded by the expectation of a certain random variable on the set of natural numbers. Then, they identify a large class of weakly renormalizable series of Dirichlet type, whose analysis depends on the properties of a difference operator called Bernoulli operator. It is also shown that the probabilistic renormalization is compatible with analytic continuation. The main results of the paper are the following:
If \(s\neq 0\), the series corresponding to \(\mathbf{a}=(a_n)_{n=0}^\infty\) is weakly renormalizable and \[ \mathbb{E}[X_{\mathbf{a}}]=-\frac{1}{s}\left(\mathbf{B}f(s,t_0)-m^s\mathbf{B}f^m\left(s,\frac{t_0-1}{m}+1\right)\right). \]
The value of \(\mathbb{E}[X_{\mathbf{a}}]\) can be expressed in terms of the analytic continuation of the Dirichlet series associated with \(f(s,t)\): \[ \mathbb{E}[X_{\mathbf{a}}]=\mathbf{D}^f(1-s,t_0)-\mathbf{D}^{\mathbf{S}_mf}\left(1-s,\frac{t_0+m-1}{m}\right). \]
Let \(s\neq 1\). The series associated to the sequence \(\mathbf{a}\) defined by \(a_0=0\), \({a_n=1/n^s}\), \(n\geq 1\), is strongly renormalizable and \[ \mathbb{E}[X_{\mathbf{a}}]=(1-m^s)\zeta(s), \] where \(\zeta(s)\) is the Riemann zeta function.
The complex vector space consisting of continuous \(s\)-Kubert functions is two-dimensional and consists of real analytic functions. Actually, it is spanned by \(-s\zeta(1-s,t)\) and \(\ell(s,t)\).

MSC:

11M41 Other Dirichlet series and zeta functions
40A05 Convergence and divergence of series and sequences
40G99 Special methods of summability
30B40 Analytic continuation of functions of one complex variable
30B50 Dirichlet series, exponential series and other series in one complex variable
28A12 Contents, measures, outer measures, capacities
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