Alejandro Malla - cp's OEIS frontend

User: Alejandro Malla

Alejandro Malla has authored 2 sequences.

A367309 Decimal expansion of area under the curve (1-2^(1-x))*zeta(x) from 0 to 1.

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

Offset: 0

Alejandro Malla, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to g(x) = (1 - 2^(1-x))*zeta(x) on the open interval (0, 1). This series can be described as an alternating version of the 'p-series' when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x. Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined, but has the limit value log(2). Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			0.60211234931037155497112632...

Cf. A113024, A367310, A367311.

y = NIntegrate[(1 - 2^(1-x)) Zeta[x], {x, 0, 1}, WorkingPrecision -> 200]
RealDigits[y][[1]]

intnum(x=0, 1, (1-2^(1-x))*zeta(x)) \\ Michel Marcus, Nov 14 2023

A367310 Continued fraction of the constant in A367309 (the area under the curve (1 - 2^(1-x)) zeta(x) from 0 to 1).

Original entry on oeis.org

0, 1, 1, 1, 1, 18, 2, 1, 37, 3, 1, 2, 20, 1, 3, 3, 1, 1, 1, 1, 2, 1, 26, 1, 1, 1, 2, 4, 3, 1, 76, 1, 3, 1, 1, 2, 1, 1, 3, 5, 3, 1, 10, 2, 5, 1, 3, 8, 1, 3, 2, 2, 5, 2, 1, 1, 1, 6, 2, 36, 5, 1, 2, 1, 1, 1, 9, 4, 3, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 29, 1

Offset: 0

Alejandro Malla, Nov 13 2023

Cf. A367309, A367311.

ContinuedFraction[NIntegrate[(1 - 2^(1-x)) Zeta[x], {x,0,1}, WorkingPrecision ->300]]

localprec(100); contfrac(intnum(x=0, 1, (1-2^(1-x))*zeta(x))) \\ Michel Marcus, Nov 14 2023

cp's OEIS Frontend

User: Alejandro Malla

A367309 Decimal expansion of area under the curve (1-2^(1-x))*zeta(x) from 0 to 1.

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