Asymptotic Approximations

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Asymptotic Approximations

\RiemannZeta ⁢ @ ⁢ σ + i ⁢ t = ∑ 1 ≤ n ≤ x 1 n s + χ ⁢ ( s ) ⁢ ∑ 1 ≤ n ≤ y 1 n 1 - s + \BigO ⁢ @ ⁢ x - σ + \BigO ⁢ @ ⁢ y σ - 1 ⁢ t 1 2 - σ \RiemannZeta @ 𝜎 imaginary-unit 𝑡 subscript 1 𝑛 𝑥 1 superscript 𝑛 𝑠 𝜒 𝑠 subscript 1 𝑛 𝑦 1 superscript 𝑛 1 𝑠 \BigO @ superscript 𝑥 𝜎 \BigO @ superscript 𝑦 𝜎 1 superscript 𝑡 1 2 𝜎 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{\sigma+\mathrm{i}t}=% \sum_{1\leq n\leq x}\frac{1}{n^{s}}+\chi(s)\sum_{1\leq n\leq y}\frac{1}{n^{1-s% }}+\BigO@{x^{-\sigma}}+\BigO@{y^{\sigma-1}t^{\frac{1}{2}-\sigma}}}}} {\displaystyle \RiemannZeta@{\sigma+\iunit t} = \sum_{1 \leq n \leq x} \frac{1}{n^s} + \chi(s) \sum_{1 \leq n \leq y} \frac{1}{n^{1-s}} + \BigO@{x^{-\sigma}} + \BigO@{y^{\sigma-1} t^{\frac{1}{2} - \sigma}} }

Substitution(s): s = σ + i ⁢ t 𝑠 𝜎 imaginary-unit 𝑡 {\displaystyle{\displaystyle{\displaystyle s=\sigma+\mathrm{i}t}}} &
χ ⁢ ( s ) = π s - 1 2 ⁢ Γ ⁡ ( 1 2 - 1 2 ⁢ s ) / Γ ⁡ ( 1 2 ⁢ s ) 𝜒 𝑠 superscript 𝜋 𝑠 1 2 Euler-Gamma 1 2 1 2 𝑠 Euler-Gamma 1 2 𝑠 {\displaystyle{\displaystyle{\displaystyle{\displaystyle\chi(s)=\pi^{s-\frac{1% }{2}}\Gamma\left(\tfrac{1}{2}-\tfrac{1}{2}s\right)/\Gamma\left(\tfrac{1}{2}s% \right)}}}} &
t = 2 ⁢ π ⁢ x ⁢ y 𝑡 2 𝜋 𝑥 𝑦 {\displaystyle{\displaystyle{\displaystyle t=2\pi xy}}}


Constraint(s): x ≥ 1 𝑥 1 {\displaystyle{\displaystyle{\displaystyle x\geq 1}}} &
y ≥ 1 𝑦 1 {\displaystyle{\displaystyle{\displaystyle y\geq 1}}} &
0 ≤ σ ≤ 1 0 𝜎 1 {\displaystyle{\displaystyle{\displaystyle 0\leq\sigma\leq 1}}} &
formula valid as t → ∞ → 𝑡 {\displaystyle{\displaystyle{\displaystyle t\to\infty}}} with σ 𝜎 {\displaystyle{\displaystyle{\displaystyle\sigma}}} fixed


\RiemannZeta ⁢ @ ⁢ 1 2 + i ⁢ t = ∑ n = 1 m 1 n 1 2 + i ⁢ t + χ ⁢ ( 1 2 + i ⁢ t ) ⁢ ∑ n = 1 m 1 n 1 2 - i ⁢ t + \BigO ⁢ @ ⁢ t - 1 / 4 \RiemannZeta @ 1 2 imaginary-unit 𝑡 superscript subscript 𝑛 1 𝑚 1 superscript 𝑛 1 2 imaginary-unit 𝑡 𝜒 1 2 imaginary-unit 𝑡 superscript subscript 𝑛 1 𝑚 1 superscript 𝑛 1 2 imaginary-unit 𝑡 \BigO @ superscript 𝑡 1 4 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{\tfrac{1}{2}+\mathrm{i% }t}=\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi\left(\tfrac{1}{2}% +\mathrm{i}t\right)\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+\BigO@{% t^{-1/4}}}}} {\displaystyle \RiemannZeta@{\tfrac{1}{2}+\iunit t} = \sum_{n=1}^m \frac{1}{n^{\frac{1}{2}+\iunit t}} + \chi\left( \tfrac{1}{2}+\iunit t \right) \sum_{n=1}^m \frac{1}{n^{\frac{1}{2}-\iunit t}} + \BigO@{t^{-1/4}} }

Constraint(s): formula valid as t → ∞ → 𝑡 {\displaystyle{\displaystyle{\displaystyle t\to\infty}}}


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