Artur Jasinski - cp's OEIS frontend

A387303 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(4n)/(4n), where P(x) is the prime zeta function.

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

Offset: 0

Artur Jasinski, Aug 25 2025

			0.0187602016899806686319119770449145568...

Cf. A387289, A387293, A387300.

RealDigits[Log[105^(1/4)/Pi], 10, 105, -1][[1]]

Equals log(105^(1/4)/Pi).

For m > 1, Sum_{k>=1} (-1)^(k+1) * primezeta(m*k)/k = log(zeta(m)/zeta(2*m)). - Vaclav Kotesovec, Aug 25 2025

A387300 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(2n)/(2n), where P(x) is the prime zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 25 2025

			0.20929521470170485885457493372...

Cf. A387289, A387293, A387303.

RealDigits[Log[Sqrt[15]/Pi], 10, 105][[1]]

Equals log(sqrt(15)/Pi).

For m > 1, Sum_{k>=1} (-1)^(k+1) * primezeta(m*k)/k = log(zeta(m)/zeta(2*m)). - Vaclav Kotesovec, Aug 25 2025

A387289 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(3n)/(3n), where P(x) is the prime zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 25 2025

			0.055613262596277710178747463453...

Cf. A387293.

RealDigits[Log[Zeta[3]/Zeta[6]]/3, 10, 105, -1][[1]]

Equals log(zeta(3)/zeta(6))/3.
Equals log(3*(35*zeta(3))^(1/3)/Pi^2).
Sum_{p prime} Sum_{n>=1} (-1)^(n+1)/p^(3*n)/(3*n) = Sum_{p prime} log((1+1/p^3))/3 = log(Product_{p prime} (1+1/p^3))/3 = log(zeta(3)/zeta(6))/3. - Amiram Eldar, Aug 25 2025

A387293 Decimal expansion of Sum_{k>=2} P(k)/zeta(k), where P(k) is the prime zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 25 2025

			0.5589554346445038949890731220538433207896034...

Cf. A387289.

A387065 Decimal expansion of the value of the Riemann zeta function at 2nd extremum (negated).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 15 2025

			-0.00398644166367075043171...

Artur Jasinski, Values of Riemann zeta function at 1000 succesive negative extrema.

Cf. A271855, A271856, A387052.

kk = x /. FindRoot[Zeta'[x] == 0, {x, -5}, WorkingPrecision -> 110]; dd =
 RealDigits[Zeta[kk], 10, 105][[1]]; Flatten[Prepend[dd, {0, 0}]]

Equals zeta(-A387052).

A387052 Decimal expansion of -x, where x is the abscissa of the second local extremum of the Riemann zeta function on the negative real axis.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Aug 15 2025

The Riemann zeta function has zeros for x = -2*n for n >= 1, which means that between -2*(n+1) and -2*n the function has an extremum for each positive integer n.
For the value of zeta(-x_2) see A387065.
It is an open question whether the fractional part of x_n tends to 1 or some unknown constant c < 1 as n tends to infinity.

Artur Jasinski, Complete list of the first 1000 negative extrema of the Riemann zeta function.

Cf. A271855, A271856.

kk = x /. FindRoot[Zeta'[x] == 0, {x, -5}, WorkingPrecision -> 110];
RealDigits[kk, 10, 105][[1]]

PARI
```
solve(x=-4.95, -4.9, zeta'(x))
```

A383903 Decimal expansion of Meissel Prime theta function at x = 2 : Sum_{p prime} 1/exp(2*p).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 07 2025

Meissel Prime theta function is defined : Sum_{p prime} 1/exp(x*p).

			0.02084062280793977076142879543476453588872816...

Ernst Meissel, Bericht über die Provinzial-Gewerbe-Schule zu Iserlohn. Notiz No. 38 pp.1-17 (manuscript).

Peter Lindqvist and Jaak Peetre, On a number theoretic sum considered by Meissel : a historical observation, Nieuw Archief voor Wiskunde (Serie 4) 1997 Vol. 15 (3) pp. 175-179 (see reprint p. 3).

Cf. A077761, A383901.

evalf[140](sum(1/exp(2*ithprime(i)), i=1..infinity));  # Alois P. Heinz, Aug 07 2025

sum = 0; Do[sum = sum + N[1/E^(2 Prime[n]), 110], {n, 1, 56}];
RealDigits[sum, 10, 105][[1]]

sumpos(k = 1, exp(-2*prime(k))) \\ Amiram Eldar, Aug 08 2025

A383901 Decimal expansion of Meissel Prime theta function at 1 : Sum_{p prime} 1/exp(p).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 07 2025

Meissel Prime theta function is defined : Sum_{p prime} 1/exp(x*p).

			0.192791189704395145042150125418556376792365...

Ernst Meissel, Bericht über die Provinzial-Gewerbe-Schule zu Iserlohn. Notiz No. 38 pp. 1-17 (manuscript).

Peter Lindqvist and Jaak Peetre, On a number theoretic sum considered by Meissel : a historical observation, Nieuw Archief voor Wiskunde (Serie 4) 1997 Vol. 15 (3) pp. 175-179 (see reprint p. 3).

Cf. A077761, A383903.

evalf[140](sum(1/exp(ithprime(i)), i=1..infinity));  # Alois P. Heinz, Aug 07 2025

sum = 0; Do[sum = sum + N[1/E^(Prime[n]), 110], {n, 1, 56}];
RealDigits[sum, 10, 105][[1]]

sumpos(k = 1, exp(-prime(k))) \\ Amiram Eldar, Aug 08 2025

A386744 Decimal expansion of the Product {n>=2, Omega(n)=2} 1/(1 - 1/n^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Aug 01 2025

			1.154135429131192212753136476082653062021377019769166311601...

Richard Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011, Table 2 p. 4.

Cf. A001222 (Omega), A001358.

A385423 Decimal expansion of the absolute value of zeta(1/5).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jul 26 2025

			0.7339209248963405922438096137551368666491286938...

Cf. A251734, A386590.

evalf(abs(Zeta(1/5)), 120);  # Alois P. Heinz, Jul 26 2025

Mathematica
```
RealDigits[-Zeta[1/5], 10, 105][[1]]
```

Equals (1/(2^(4/5)-1))*Sum_{k>=1} (-1)^(k+1)/k^(1/5).

cp's OEIS Frontend

User: Artur Jasinski

A387303 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(4*n)/(4*n), where P(x) is the prime zeta function.

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A387300 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(2*n)/(2*n), where P(x) is the prime zeta function.

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A387289 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(3*n)/(3*n), where P(x) is the prime zeta function.

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A387293 Decimal expansion of Sum_{k>=2} P(k)/zeta(k), where P(k) is the prime zeta function.

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A387065 Decimal expansion of the value of the Riemann zeta function at 2nd extremum (negated).

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A387052 Decimal expansion of -x, where x is the abscissa of the second local extremum of the Riemann zeta function on the negative real axis.

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A383903 Decimal expansion of Meissel Prime theta function at x = 2 : Sum_{p prime} 1/exp(2*p).

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A383901 Decimal expansion of Meissel Prime theta function at 1 : Sum_{p prime} 1/exp(p).

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A387303 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(4n)/(4n), where P(x) is the prime zeta function.

A387300 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(2n)/(2n), where P(x) is the prime zeta function.

A387289 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(3n)/(3n), where P(x) is the prime zeta function.