# Infinite double sum with exclusion

I would like to evaluate the following sum (numerically will suffice): $$\sum_{m,n=-\infty,(m,n)\neq(0,0)}^{\infty}\frac{1}{m^{4}+n^{4}}$$ I first tried to do $$\sum_{m,n=1}^{\infty}\frac{1}{m^{4}+n^{4}}$$

NSum[1/(m^4 + n^4), {m, 1, Infinity}, {n, 1, Infinity}]


but got the error "Summand (or its derivative) -((4.\m^3)/(m^4+n^4)^2) is not numerical
at point n = 1.." I don't understand this as it's a well-behaved, convergent sum. What is going on here?

In terms of the original sum I want to calculate, I'm not sure how to elegantly eliminate the $$(m,n)=(0,0)$$ term. I thought of using a Kronecker Delta like

NSum[(1 - KroneckerDelta[m, 0] KroneckerDelta[n, 0])/(m^4 +
n^4), {m, -Infinity, Infinity}, {n, -Infinity, Infinity}]


but it still complains about the infinite term. How can I avoid this?

Thanks in advance for any help.

• For the first sum, you can exclude $(0,0)$ by making it contribute zero to the sum at that point with Piecewise[{{0, m == n == 0}}, 1/(m^4 + n^4)] Apr 26, 2021 at 16:41
• A similar question was asked, answered, and closed here. Apr 26, 2021 at 17:44

Too long for a comment. The double sum under consideration can be treated as an iterated sum and the inner sum has a closed form:

f[n_] := Sum[Piecewise[{{1/(m^4 + n^4), m^2 + n^2 != 0}, {0,
True}}], {m, -Infinity, Infinity}]
a=Assuming[n\[Element]PositiveIntegers,f[n]]//ComplexExpand


-((\[Pi] Sin[Sqrt n \[Pi]])/( Sqrt n^3 (Cos[Sqrt n \[Pi]] - Cosh[Sqrt n \[Pi]]))) - (\[Pi] Sinh[Sqrt n \[Pi]])/( Sqrt n^3 (Cos[Sqrt n \[Pi]] - Cosh[Sqrt n \[Pi]]))

or in $$\LaTeX$$ $$-\frac{\pi \sin \left(\sqrt{2} \pi n\right)}{\sqrt{2} n^3 \left(\cos \left(\sqrt{2} \pi n\right)-\cosh \left(\sqrt{2} \pi n\right)\right)}-\frac{\pi \sinh \left(\sqrt{2} \pi n\right)}{\sqrt{2} n^3 \left(\cos \left(\sqrt{2} \pi n\right)-\cosh \left(\sqrt{2} \pi n\right)\right)}$$ and f==\[Pi]^4/45. You may play with it on your own. It's time to go to bed for me.

Divide the index ranges from - Infinity to Infinity into separate sums from - Inf to - 1 and 1 to Inf plus index = 0 and you get partial sums (1/(m^4 + n^4) is even) you can evaluate like

Table[4 (ParallelSum[1./(m^4 + n^4), {m, 1, 10^k}, {n, 1, 10^k}] +
ParallelSum[1./(m^4), {m, 1, 10^k}]), {k, 1, 4}]
(* {7.34484946634854, 7.375876361817944, 7.3762162344005695, 7.376219664150702} *)


With index pairs {m,n} shown in the plot the sum is easy evaluated to

4 NSum[1/((m - n)^4 + n^4), {m, 1, Infinity}, {n, 0, m - 1}]
(*7.37602*)

• Why 7.376022908995918 < 7.376219664150702 as in the Andreas' answer? Apr 27, 2021 at 16:00
• @user64494 compare 4*NSum[1/((m - n)^4 + n^4), {m, 1, 10^3}, {n, 0, m - 1}] with 4*N[Sum[1/((m - n)^4 + n^4), {m, 1, 10^3}, {n, 0, m - 1}], 10]`.The sums increase monotonically with the stop summation index. To use NSum with large summation index (or infinity) may be calling for trouble. Apr 27, 2021 at 17:06