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Fungrim entry: 792f7b

ζ ⁣(s)=k=1N11ks+N1ss1+1Ns(12+k=1MB2k(2k)!(s)2k1N2k1)NB2M ⁣(tt)(2M)!(s)2Mts+2Mdt\zeta\!\left(s\right) = \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) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{t}^{s + 2 M}} \, dt
Assumptions:sCands1andNZandMZandRe ⁣(s+2M1)>0andN1andM1s \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:
\zeta\!\left(s\right) = \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) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{t}^{s + 2 M}} \, dt

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
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
BernoulliBBnB_{n} Bernoulli number
Factorialn!n ! Factorial
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Integralabf ⁣(x)dx\int_{a}^{b} f\!\left(x\right) \, dx Integral
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("792f7b"),
    Formula(Equal(RiemannZeta(s), Sub(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))))), Integral(Mul(Div(BernoulliPolynomial(Mul(2, M), Sub(t, Floor(t))), Factorial(Mul(2, M))), Div(RisingFactorial(s, Mul(2, M)), Pow(t, Add(s, Mul(2, M))))), Tuple(t, N, Infinity))))),
    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."))

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2019-08-17 11:32:46.829430 UTC