# Problem with a finite sum involving HurwitzZeta Function

I'm trying to reproduce some plots from the analytical expression:

$$f(\xi)=\left(\frac{2}{\beta^2}-1\right)+\left(\frac{\theta\,e^{-\beta}}{2}-\frac{1}{2}\right)\xi+\sum_{n=1}^{60}\left[\frac{(-\beta)^n}{n!}\xi^{1+n/2}\zeta_H\left(-\frac{n}{2},1+\frac{1}{\xi}\right)\right]$$.

I need to plot $$f(-4x)$$ for several values of $$\beta$$ and with $$\theta=0$$ and $$\theta=1$$. In the attached picture, the left panel is what I have to get (ignore the blue curves), but Mathematica gives me a lot of noise.

My code is:

f[β_, ξ_,M_, θ_] := ((2/β^2 -1) + ((θ Exp[-β])/2 - 1/2) ξ + Sum[((-β)^n ξ^(1 + n/2))/n! HurwitzZeta[-(n/2), 1 + 1/ξ], {n, 1, M}]);

Plot[{f[0.5, -4 x, 60, 1], f[0.5, 4 x, 60, 0],
f[0.8, -4 x, 60, 1], f[0.8, 4 x, 60, 0], f[1.1, -4 x, 60, 1],
f[1.1, 4 x, 60, 0]}, {x, -6, 6}, AxesOrigin -> {0, 0},
PlotStyle -> {Green, Green, Red, Red, Black, Black},
PlotRange -> {0, 8}]


Do you have an idea of what is going on?

Thanks!

• For now, try using WorkingPrecision -> 30 in Plot[]. May 31, 2020 at 2:15

Clear["Global*"]

f[β_, ξ_, M_, θ_] := ((2/β^2 - 1)
+ ((θ Exp[-β])/2 - 1/2) ξ +
Sum[((-β)^n ξ^(1 + n/2))/n! HurwitzZeta[-(n/2),
1 + 1/ξ], {n, 1, M}]);


To use high precision, make the input parameters exact values.

Plot[{
f[1/2, -4 x, 60, 1],
f[1/2, 4 x, 60, 0],
f[4/5, -4 x, 60, 1],
f[4/5, 4 x, 60, 0],
f[11/10, -4 x, 60, 1],
f[11/10, 4 x, 60, 0]},
{x, -6, 6},
AxesOrigin -> {0, 0},
Frame -> True,
PlotStyle -> {Green, {Green, Dashed}, Red, {Red, Dashed},
Black, {Black, Dashed}},
PlotRange -> {0, 8},
WorkingPrecision -> 50,
PlotPoints -> 100,
MaxRecursion -> 5,
PlotLegends -> "Expressions"]
`

Just change the styles back to get rid of the dashes.

• Thank you all guys. It solved my problem. :) May 31, 2020 at 19:17