Fungrim entry: e37535

\left|\left(1 - {2}^{1 - s}\right) \zeta\!\left(s\right) - \frac{1}{d(n)} \sum_{k=0}^{n - 1} \frac{{\left(-1\right)}^{k} \left(d(n) - d(k)\right)}{{\left(k + 1\right)}^{s}}\right| \le \frac{3 \left(1 + 2 \left|\operatorname{Im}(s)\right|\right)}{{\left(3 + \sqrt{8}\right)}^{n}} {e}^{\left|\operatorname{Im}(s)\right| \pi / 2}\; \text{ where } d(k) = n \sum_{i=0}^{k} \frac{\left(n + i - 1\right)! {4}^{i}}{\left(n - i\right)! \left(2 i\right)!}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) \ge \frac{1}{2} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

References:

P. Borwein. An efficient algorithm for the Riemann zeta function. Canadian Mathematical Society Conference Proceedings, vol. 27, pp. 29-34. 2000.

TeX:

\left|\left(1 - {2}^{1 - s}\right) \zeta\!\left(s\right) - \frac{1}{d(n)} \sum_{k=0}^{n - 1} \frac{{\left(-1\right)}^{k} \left(d(n) - d(k)\right)}{{\left(k + 1\right)}^{s}}\right| \le \frac{3 \left(1 + 2 \left|\operatorname{Im}(s)\right|\right)}{{\left(3 + \sqrt{8}\right)}^{n}} {e}^{\left|\operatorname{Im}(s)\right| \pi / 2}\; \text{ where } d(k) = n \sum_{i=0}^{k} \frac{\left(n + i - 1\right)! {4}^{i}}{\left(n - i\right)! \left(2 i\right)!}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) \ge \frac{1}{2} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Pow`	${a}^{b}$	Power
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Sqrt`	$\sqrt{z}$	Principal square root
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`Factorial`	$n !$	Factorial
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e37535"),
    Formula(Where(LessEqual(Abs(Sub(Mul(Sub(1, Pow(2, Sub(1, s))), RiemannZeta(s)), Mul(Div(1, d(n)), Sum(Div(Mul(Pow(-1, k), Sub(d(n), d(k))), Pow(Add(k, 1), s)), For(k, 0, Sub(n, 1)))))), Mul(Div(Mul(3, Add(1, Mul(2, Abs(Im(s))))), Pow(Add(3, Sqrt(8)), n)), Exp(Div(Mul(Abs(Im(s)), Pi), 2)))), Equal(d(k), Mul(n, Sum(Div(Mul(Factorial(Sub(Add(n, i), 1)), Pow(4, i)), Mul(Factorial(Sub(n, i)), Factorial(Mul(2, i)))), For(i, 0, k)))))),
    Variables(s, n),
    Assumptions(And(Element(s, CC), GreaterEqual(Re(s), Div(1, 2)), NotEqual(s, 1), Element(n, ZZGreaterEqual(1)))),
    References("P. Borwein. An efficient algorithm for the Riemann zeta function. Canadian Mathematical Society Conference Proceedings, vol. 27, pp. 29-34. 2000."))

Topics using this entry

Riemann zeta function