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• W2300919129 abstract "We present several formulae for the large <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> asymptotics of the Riemann zeta function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=zeta left-parenthesis s right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ζ<!-- ζ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>zeta (s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s equals sigma plus i t> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>s=sigma +i t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 less-than-or-equal-to sigma less-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0leq sigma leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>t>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation Underscript a Overscript b Endscripts n Superscript negative s> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:munderover> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>sum _a^b n^{-s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for certain ranges of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=application/x-tex>b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In addition, we present precise estimates relating this sum with the sum <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation Underscript c Overscript d Endscripts n Superscript s minus 1> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mi>c</mml:mi> <mml:mi>d</mml:mi> </mml:munderover> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>s</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>sum _c^d n^{s-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for certain ranges of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a comma b comma c comma d> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>a, b, c, d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also study a two-parameter generalization of the Riemann zeta function which we denote by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Phi left-parenthesis u comma v comma beta right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Phi (u,v,beta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u element-of double-struck upper C> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>uin mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=v element-of double-struck upper C> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>vin mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta element-of double-struck upper R> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>beta in mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Generalizing the methodology used in the study of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=zeta left-parenthesis s right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ζ<!-- ζ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>zeta (s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we derive asymptotic formulae for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Phi left-parenthesis u comma v comma beta right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Phi (u,v, beta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2300919129 created "2016-06-24" @default.
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• W2300919129 date "2022-01-01" @default.
• W2300919129 modified "2023-09-30" @default.
• W2300919129 title "On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function" @default.
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