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).

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

Offset

0,6

Comments

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

Links

Table of n, a(n) for n=0..20.

See A332645.

Formula

No explicit formula is known.

Example

0.000004438269312506953

Mathematica

(* 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]

Crossrefs

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

Sequence in context: A258075 A286296 A023530 * A345294 A233581 A193628

Adjacent sequences: A337362 A337363 A337364 * A337366 A337367 A337368

Keyword

nonn,cons,more

Author

Artur Jasinski, Aug 26 2020

Status

approved