Fungrim entry: d31b04

\left|\zeta\!\left(s\right) - \left(\sum_{k=1}^{N - 1} \frac{1}{{k}^{s}} + \frac{{N}^{1 - s}}{s - 1} + \frac{1}{{N}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{N}^{2 k - 1}}\right)\right)\right| \le \frac{4 \left|\left(s\right)_{2 M}\right|}{{\left(2 \pi\right)}^{2 M}} \frac{{N}^{-\left(\operatorname{Re}\!\left(s\right) + 2 M - 1\right)}}{\operatorname{Re}\!\left(s\right) + 2 M - 1}

Assumptions:

s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, s \ne 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s + 2 M - 1\right) \gt 0 \,\mathbin{\operatorname{and}}\, N \ge 1 \,\mathbin{\operatorname{and}}\, M \ge 1

References:

F. Johansson (2015), Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, Numerical Algorithms 69:253, DOI: 10.1007/s11075-014-9893-1
F. W. J. Olver, Asymptotics and Special Functions, AK Peters, 1997. Chapter 8.

TeX:

\left|\zeta\!\left(s\right) - \left(\sum_{k=1}^{N - 1} \frac{1}{{k}^{s}} + \frac{{N}^{1 - s}}{s - 1} + \frac{1}{{N}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{N}^{2 k - 1}}\right)\right)\right| \le \frac{4 \left|\left(s\right)_{2 M}\right|}{{\left(2 \pi\right)}^{2 M}} \frac{{N}^{-\left(\operatorname{Re}\!\left(s\right) + 2 M - 1\right)}}{\operatorname{Re}\!\left(s\right) + 2 M - 1}

s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, s \ne 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s + 2 M - 1\right) \gt 0 \,\mathbin{\operatorname{and}}\, N \ge 1 \,\mathbin{\operatorname{and}}\, M \ge 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Pow`	${a}^{b}$	Power
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`ConstPi`	$\pi$	The constant pi (3.14...)
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d31b04"),
    Formula(LessEqual(Abs(Sub(RiemannZeta(s), Parentheses(Add(Add(Sum(Div(1, Pow(k, s)), Tuple(k, 1, Sub(N, 1))), Div(Pow(N, Sub(1, s)), Sub(s, 1))), Mul(Div(1, Pow(N, s)), Add(Div(1, 2), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Div(RisingFactorial(s, Sub(Mul(2, k), 1)), Pow(N, Sub(Mul(2, k), 1)))), Tuple(k, 1, M)))))))), Mul(Div(Mul(4, Abs(RisingFactorial(s, Mul(2, M)))), Pow(Mul(2, ConstPi), Mul(2, M))), Div(Pow(N, Neg(Parentheses(Sub(Add(Re(s), Mul(2, M)), 1)))), Sub(Add(Re(s), Mul(2, M)), 1))))),
    Assumptions(And(Element(s, CC), Unequal(s, 1), Element(N, ZZ), Element(M, ZZ), Greater(Re(Sub(Add(s, Mul(2, M)), 1)), 0), GreaterEqual(N, 1), GreaterEqual(M, 1))),
    Variables(s, N, M),
    References("F. Johansson (2015), Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, Numerical Algorithms 69:253, DOI: 10.1007/s11075-014-9893-1", "F. W. J. Olver, Asymptotics and Special Functions, AK Peters, 1997. Chapter 8."))

Topics using this entry

Riemann zeta function