Transcendence of power series for some number theoretic functions

<p>We give a new proof of Fatou’s theorem: <italic>if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.</italic> This result is applied to show that for any non–trivial completely multiplicative function from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet negative 1 comma 1 EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{-1,1\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the series <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts f left-parenthesis n right-parenthesis z Superscript n"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:munderover> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{n=1}^\infty f(n)z^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is transcendental over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; in particular, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts lamda left-parenthesis n right-parenthesis z Superscript n"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:munderover> <mml:mi>λ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{n=1}^\infty \lambda (n)z^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is transcendental, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi>λ</mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Liouville’s function. The transcendence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts mu left-parenthesis n right-parenthesis z Superscript n"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:munderover> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{n=1}^\infty \mu (n)z^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also proved ...