Fungrim entry: 792f7b

\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:

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:

\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) \gt 0 \,\mathbin{\operatorname{and}}\, N \ge 1 \,\mathbin{\operatorname{and}}\, M \ge 1

Definitions:

Fungrim symbol	Notation	Short description
`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
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers
`Re`	$\operatorname{Re}\!\left(z\right)$	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."))

Topics using this entry

Riemann zeta function