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Fungrim entry: d31b04

ζ ⁣(s)(k=1N11ks+N1ss1+1Ns(12+k=1MB2k(2k)!(s)2k1N2k1))4(s)2M(2π)2MN(Re(s)+2M1)Re(s)+2M1\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}(s) + 2 M - 1\right)}}{\operatorname{Re}(s) + 2 M - 1}
Assumptions:sC  and  s1  and  NZ  and  MZ  and  Re ⁣(s+2M1)>0  and  N1  and  M1s \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) > 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}(s) + 2 M - 1\right)}}{\operatorname{Re}(s) + 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) > 0 \;\mathbin{\operatorname{and}}\; N \ge 1 \;\mathbin{\operatorname{and}}\; M \ge 1
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
BernoulliBBnB_{n} Bernoulli number
Factorialn!n ! Factorial
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Piπ\pi The constant pi (3.14...)
ReRe(z)\operatorname{Re}(z) Real part
CCC\mathbb{C} Complex numbers
ZZZ\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)), For(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)))), For(k, 1, M)))))))), Mul(Div(Mul(4, Abs(RisingFactorial(s, Mul(2, M)))), Pow(Mul(2, Pi), 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), NotEqual(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."))

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2021-03-15 19:12:00.328586 UTC