A104540 -id:A104540 - cp's OEIS frontend

A335815 Decimal expansion of Sum_{n>=1} 1/z(n)^4 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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

Offset: 0

Artur Jasinski, Jun 25 2020

Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; this sequence.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.

			0.0000371725992852696861648662624717405784536508897300...

André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.25 Table 2.
André Voros, Zeta functions for the Riemann zeros, 2001(2008) p.20 Table 1.
André Voros, Zeta functions for the Riemann zeros, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, p. 665-699.
André Voros, Zeta functions over Zeros of the Zeta functions, 2010, p. 153.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814.

Join[{0,0,0,0},RealDigits[N[-1/12*(D[Log[Zeta[x]],{x,4}]/. x -> 1/2) - 1/24 Pi^4 -(Zeta[4, 1/4] - Zeta[4, 3/4])/64 + 16, 105]][[1]]]

Equals 16-Pi^4/24+(Zeta[4,3/4]-Zeta[4,1/4])/64-(Log[Zeta[x]]''''[1/2])/24

A335826 Decimal expansion of Sum_{n>=1} 1/z(n)^6 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jun 25 2020

Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.

			0.000000144173931400973279695381556....

André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.25 Table 2.
André Voros, Zeta functions for the Riemann zeros, 2001(2008) p.20 Table 1.
André Voros, Zeta functions for the Riemann zeros, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, p. 665-699.
André Voros, Zeta functions over Zeros of the Zeta functions, 2010, p. 153.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815.

m = 3; Join[{0, 0, 0, 0, 0, 0},RealDigits[N[((-1)^m (2^(2 m) - ((2^(2 m) - 1) Zeta[2 m] + (Zeta[2 m, 1/4] - Zeta[2 m, 3/4])/2^(2 m))/4 - (D[Log[Zeta[x]], {x, 2 m}] /. x -> 1/2)/(2 (2 m - 1)!) )), 105]][[1]]]

Universal formula for Sum_{n>=1} 1/z(n)^(2m) published in Voros 2002-2003 p. 22 (see Mathematica procedure below).

A355283 Decimal expansion of the constant B(3) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^3 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 20 2022

			0.0000001940333754063698367... = 1.940333754063698367*10^(-7).

For links see A332645.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815.

Equals (A333360^2 - A335826)/2.

No simpler formula is known.

A246843 Decimal expansion of C, a constant associated with the estimation of the maximum of |zeta(1+i*t)|.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 05 2014

			-0.089326522343551...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.
Tadej Kotnik, Computational estimation of the constant β(1) characterizing the order of ζ(1 + it), Math. Comp. 77: 1713-1723, 2008.

Cf. A074760, A104539, A104540, A242056.

digits = 15; precision = 200; u0 = 10^8; du = 10^8; tail[u_] := -(1 + Log[2*Pi*u])/(2*u); Clear[f]; f[u_] := f[u] = 1 - Log[2] + NIntegrate[Log[BesselI[0, t]]/t^2, {t, 0, 2} , WorkingPrecision -> precision] + NIntegrate[(Log[BesselI[0, t]] - t)/t^2, {t, 2, u}, WorkingPrecision -> precision, MaxRecursion -> 20 ] + tail[u]; f[u0]; f[u = u0 + du]; While[RealDigits[f[u], 10, digits + 4] != RealDigits[f[u - du], 10, digits + 4], Print["u = ", u, " ", f[u]]; u = u + du]; Join[{0}, RealDigits[f[u], 10, digits] // First]

1 - log(2) + integral_{0..2} log(BesselI(0, t))/t^2 dt + integral_{2..infinity} (log(BesselI(0, t)) - t)/t^2 dt.

Typo in the formula corrected by Vaclav Kotesovec, Sep 17 2014

A337365 Decimal expansion of imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 26 2020

For the decimal expansion of the real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function see A337404.
Sum_{m>=1} 1/(1/2 + i*z(m))^1 = 0.01154785448306... - i*A where 0.01154785448306 = A074760/2 and A > 10.5.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 = -0.0230771586479... - i*0.000728434... where -0.0230771586479 = A245275/2
Sum_{m>=1} 1/(1/2 + i*z(m))^3 = -0.000055579115726... + i*0.0007262105... where -0.000055579115726 = A245276/2
Sum_{m>=1} 1/(1/2 + i*z(m))^4 = 0.0000368136106308... + i*0.0000044382...
Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
Sum_{m>=1} 1/z(m)^6 = 0.0000001441739314...; see A335826.
Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276

			0.000004438269312506953

See A332645.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A335826.

(* 7-day-long procedure *)
kk = 0; Do[kk = kk + 1/(N[ZetaZero[n], 100])^4 , {n, 1, 1000000}]; Take[Join[{0, 0, 0, 0, 0}, RealDigits[Im[kk]][[1]]], 11]

No explicit formula is known.

A337404 Decimal expansion of real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 26 2020

For the decimal expansion of the imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function see A337365.

A335918 Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2)^2 where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jun 29 2020

Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.

			0.0000371006364374648715125054339132797135962919799565287...

André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.22 Table 1.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815.

Join[{0, 0, 0, 0},RealDigits[N[3 + EulerGamma + EulerGamma^2 - Pi^2/8 - Log[4 Pi] + 2 StieltjesGamma[1], 105]][[1]]]

Equals: 3 + gamma + gamma^2 - Pi^2/8 - log(4*Pi) + 2*gamma(1), where gamma is the Euler-Mascheroni gamma constant (see A001620) and gamma(1) is 1st Stieltjes constant (see A082633).

A356693 Decimal expansion of the constant B(2) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 23 2022

			0.000248334053789144...

Cf. A355283 (B(3)).

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A355283.

Join[{0, 0, 0}, RealDigits[N[-4*Catalan + Catalan^2/2 - Pi^2/2 + (Catalan*Pi^2)/8 + Pi^4/128 + (1/64)*Zeta[4, 1/4] + (2*Zeta'[1/2]^2)/Zeta[1/2]^2 - (Catalan Zeta'[1/2]^2)/(2 Zeta[1/2]^2) - (Pi^2 Zeta'[1/2]^2)/(16*Zeta[1/2]^2) - Zeta'[1/2]^4/(8*Zeta[1/2]^4) - (2 Zeta''[1/2])/Zeta[1/2] + (Catalan Zeta''[1/2])/(2 Zeta[1/2]) + (Pi^2 Zeta''[1/2])/(16*Zeta[1/2]) + Zeta'[1/2]^2*Zeta''[1/2]/(4 Zeta[1/2]^3) - Zeta'[1/2] Zeta'''[1/2]/(6 Zeta[1/2]^2) + Zeta''''[1/2]/(24  Zeta[1/2]), 115]][[1]]]

Equals (A332645^2 - A335815)/2.

A360807 Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2) where z(m) is the imaginary part of the m-th nontrivial zero of the Dirichlet beta function whose real part is 1/2.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Feb 21 2023

Conjecture: Nontrivial zeros whose real part is not 1/2 do not exist.

			0.077783989961792964431079...

Kano Kono, Summary Dirichlet beta function, Alien's Mathematics, p. 5.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360.

kk = RealDigits[N[4 Log[Gamma[3/4]] + EulerGamma/2 + Log[2] - 3 Log[Pi]/2, 115]][[1]]; Prepend[kk, 0]

4*log(gamma(3/4)) + Euler/2 + log(2) - 3*log(Pi)/2 \\ Michel Marcus, Mar 15 2023

Equals 4*log(Gamma(3/4)) + A001620/2 + log(2) - 3*log(Pi)/2.
Equals A074760 - 1 + log(4) - log(Pi) + 4*log(Gamma(3/4)).
Equals 1 - A074760 + A001620 - 2*log(Pi) + 4*log(Gamma(3/4)).
Equals 3*A074760 - 3 - A001620 + 4*log(2) + 4*log(Gamma(3/4)).

cp's OEIS Frontend

A335815 Decimal expansion of Sum_{n>=1} 1/z(n)^4 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A335826 Decimal expansion of Sum_{n>=1} 1/z(n)^6 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A355283 Decimal expansion of the constant B(3) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^3 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A246843 Decimal expansion of C, a constant associated with the estimation of the maximum of |zeta(1+i*t)|.

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A337365 Decimal expansion of imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

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A337404 Decimal expansion of real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

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A335918 Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2)^2 where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function.

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A356693 Decimal expansion of the constant B(2) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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