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- (*Mathematica start*)
- (*korrekt plot av integralen*)
- (*k and q are the crucial parameters in the integrate Euler Maclaurin \
- formula for the Riemann zeta function*)
- Clear[q, k, r, m, i, s]
- Clear[q, k, r, m, i, s, n, x, a, b, t]
- k = 30;
- q = 10;
- Plot[Re[-I*((0*1/2 + I*t) - Sum[n^(-(1/2 + I*t))/Log[n], {n, 2, k}] +
- ExpIntegralEi[-(-1 + (1/2 + I*t)) Log[k]] +
- k^-(1/2 + I*t)/(2 Log[k]) +
- Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
- k^(-(1/2 + I*t) - 2*r + 1)*(1/2 + I*t)^i*
- Sum[-(i!/m!/(Log[k]^(i + 1 - m)*(1/2 + I*t)^(i - m))), {m, 0,
- i}], {i, 1, 2*r - 1}], {r, 1,
- q - 1}]) - (-I*((0*1/2 + I*t*0)*0 -
- Sum[n^(-(1/2 + I*t*0))/Log[n], {n, 2, k}] +
- ExpIntegralEi[-(-1 + (1/2 + I*t*0)) Log[k]] +
- k^-(1/2 + I*t*0)/(2 Log[k]) +
- Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
- k^(-(1/2 + I*t*0) - 2*r + 1)*(1/2 + I*t*0)^i*
- Sum[-(i!/m!/(Log[k]^(i + 1 - m)*(1/2 + I*t*0)^(i - m))), {m,
- 0, i}], {i, 1, 2*r - 1}], {r, 1, q - 1}]))], {t, 0, 60}]
- (*31.1.2017 den här fullständigt fungerande integral numeriskt \
- korrekt*)
- Clear[q, k, r, m, i, s, n, x, a, b, t]
- k = 30;
- q = 10;
- s1 = N[1/2 + I*0, 20];
- s = s1;
- a = -I (s - Sum[n^(-s)/Log[n], {n, 2, k}] +
- ExpIntegralEi[-(-1 + s) Log[k]] + k^-s/(2 Log[k]) +
- Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
- k^(-s - 2*r + 1)*s^i*
- Sum[-(i!/m!/(Log[k]^(i + 1 - m)*s^(i - m))), {m, 0, i}], {i,
- 1, 2*r - 1}], {r, 1, q - 1}]);
- s2 = N[1/2 + I*60, 20];
- s = s2;
- b = -I (s - Sum[n^(-s)/Log[n], {n, 2, k}] +
- ExpIntegralEi[-(-1 + s) Log[k]] + k^-s/(2 Log[k]) +
- Sum[Sum[BernoulliB[2*r]/((2*r)!)*Abs[StirlingS1[2*r - 1, i]]*
- k^(-s - 2*r + 1)*s^i*
- Sum[-(i!/m!/(Log[k]^(i + 1 - m)*s^(i - m))), {m, 0, i}], {i,
- 1, 2*r - 1}], {r, 1, q - 1}]);
- (b - a)
- NIntegrate[-I*Zeta[s], {s, s1, s2}, WorkingPrecision -> 20]
- NIntegrate[Zeta[1/2 + I*t], {t, 0, 60}, WorkingPrecision -> 20]
- (*end*)
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