Formula:8296

Digital Library of Mathematical Functions ID 25.2.E8

${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\mathrm{d}x,}}$

Constraint(s)

${\displaystyle{\displaystyle\Re s>0}}$
${\displaystyle{\displaystyle\Re s>0}}$
${\displaystyle{\displaystyle N=1,2,3,\dots}}$

Symbols List

• ${\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}$: Riemann zeta function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle s}}$: complex variable
 

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