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In mathematics, the Prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for : :. == Properties == The Euler product for the Riemann zeta function ζ(''s'') implies that : which by Möbius inversion gives : When ''s'' goes to 1, we have . This is used in the definition of Dirichlet density. This gives the continuation of ''P''(''s'') to , with an infinite number of logarithmic singularities at points ''s'' where ''ns'' is a pole (only ''ns=1'')), or zero of the Riemann zeta function ζ(.). The line is a natural boundary as the singularities cluster near all points of this line. If we define a sequence : then : (Exponentiation shows that this is equivalent to Lemma 2.7 by Li.) The prime zeta function is related with the Artin's constant by : where ''L''''n'' is the ''n''th Lucas number. Specific values are: + \tfrac + \tfrac + \tfrac + \tfrac + \cdots \to \infty. || |- | 2 || || |- | 3 || || |- | 4 || || |- | 5 || || |- | 9 || || |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「prime zeta function」の詳細全文を読む スポンサード リンク
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