A355251 - OEIS

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A355251

Decimal expansion of the geometric integral of the Riemann zeta function from 1 to infinity.

6, 0, 3, 4, 9, 6, 4, 4, 1, 8, 2, 2, 3, 1, 3, 4, 8, 3, 4, 7, 0, 1, 1, 0, 0, 6, 8, 0, 5, 1, 7, 0, 2, 7, 1, 8, 9, 6, 0, 2, 3, 0, 9, 6, 3, 6, 4, 9, 4, 7, 8, 4, 3, 6, 0, 9, 6, 4, 4, 0, 4, 2, 0, 2, 1, 5, 4, 4, 8, 7, 4, 0, 2, 9, 0, 7, 4, 7, 0, 1, 0, 1, 3, 3, 7, 0, 2

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OFFSET

1,1

COMMENTS

The geometric integral of a function, f(x), from a to b is defined as lim_{dx->0} Product_{i=1..n} f(x_i)^dx, where n = (b - a)/dx and x_i is a number on the interval [a + dx*(i-1), a + dx*i].

The geometric integral can be shown to be equivalent to exp(Integral_{a..b} log(f(x)) dx).

LINKS

Iain Fox, Table of n, a(n) for n = 1..2000

Wikipedia, Product integral

FORMULA

Equals exp(Integral_{s=1..oo} log(zeta(s)) ds) = e^A188157.

EXAMPLE

Equals 6.03496441822313483470110068051702718960230963649478436096...

PROG

(PARI) exp(intnum(s=1, [oo, log(2)], log(zeta(s))))

CROSSREFS

Cf. A001113, A188157.