The Stieltjes function—definition and properties

Close to the singular point <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s equals 1"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the zeta function can be represented as a Laurent series in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis s minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(s - 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The coefficients in this series are called the Stieltjes constants, and the first ones were computed already 100 years ago. In order to investigate their somewhat unexpected behavior we have defined a related function which we call the Stieltjes function, and examined its properties.