Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 RamanujanC

Source Notebook

Evaluate Ramanujan's sum

Contributed by: Jan Mangaldan

ResourceFunction["RamanujanC"][q,n]

gives Ramanujan's sum cq(n).

Details

Integer mathematical function, suitable for both symbolic and numerical manipulation.
Ramanujan's sum is also known as von Sterneck's function.
Ramanujan's sum is given by , where k ranges over all positive integers coprime and less than or equal to q.
ResourceFunction["RamanujanC"] automatically threads over lists.

Examples

Basic Examples (2) 

Evaluate c10(5):

In[1]:=
ResourceFunction["RamanujanC"][10, 5]
Out[1]=

Plot RamanujanC for different indices:

In[2]:=
DiscretePlot[{ResourceFunction["RamanujanC"][3, n], ResourceFunction["RamanujanC"][7, n], ResourceFunction["RamanujanC"][12, n]}, {n, 50}, Sequence[ Joined -> True, PlotLegends -> { "\!\(\*SubscriptBox[\(c\), \(3\)]\)(n)", "\!\(\*SubscriptBox[\(c\), \(7\)]\)(n)", "\!\(\*SubscriptBox[\(c\), \(12\)]\)(n)"}]]
Out[2]=

Scope (2) 

Show a table of Ramanujan sums:

In[3]:=
TableForm[Array[ResourceFunction["RamanujanC"], {7, 7}], TableHeadings -> {Range[7], Range[7]}]
Out[3]=

RamanujanC threads elementwise over lists:

In[4]:=
ResourceFunction["RamanujanC"][3, {2, 4, 6}]
Out[4]=

Applications (2) 

Verify the Brauer–Rademacher identity:

In[5]:=
With[{n = 42, k = 17}, DivisorSum[n, d |-> d/EulerPhi[d] MoebiusMu[n/d], d |-> CoprimeQ[d, k]] == MoebiusMu[n]/EulerPhi[n] ResourceFunction["RamanujanC"][n, k]]
Out[5]=

Express Cyclotomic[n,x] in terms of RamanujanC and BellY:

In[6]:=
With[{n = 105}, Cyclotomic[n, x] == 1 + \!\( \*UnderoverscriptBox[\(\[Sum]\), \(r = 1\), \(EulerPhi[n]\)]\( \*FractionBox[ SuperscriptBox[\(x\), \(r\)], \(r!\)] \( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(r\)]BellY[r, k, \(-Table[\(\((j - 1)\)!\) \* InterpretationBox[ TagBox[ FrameBox[ PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.65 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["\<\"RamanujanC\"\>", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->0.9 Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], Background->RGBColor[0.968627, 0.976471, 0.984314], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 1}}, FrameStyle->RGBColor[0.831373, 0.847059, 0.85098], RoundingRadius->4], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "\"RamanujanC\""}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["RamanujanC"], BoxID -> "RamanujanC", Selectable->False][n, j], {j, 1, r - k + 1}]\)]\)\)\)]
Out[6]=

Properties and Relations (5) 

RamanujanC[q,1] is the same as MoebiusMu[q]:

In[7]:=
ResourceFunction["RamanujanC"][Range[20], 1] == MoebiusMu[Range[20]]
Out[7]=

RamanujanC[q,q] is the same as EulerPhi[q]:

In[8]:=
ResourceFunction["RamanujanC"][#, #] &[Range[20]] == EulerPhi[Range[20]]
Out[8]=

RamanujanC[q,n] is a multiplicative function with respect to its first argument:

In[9]:=
With[{p = 10, q = 13, n = 81}, ResourceFunction["RamanujanC"][p, n] ResourceFunction["RamanujanC"][ q, n] == ResourceFunction["RamanujanC"][p q, n] && CoprimeQ[p, q]]
Out[9]=

Verify the definition of RamanujanC in terms of the roots of unity:

In[10]:=
With[{q = 14, n = 8}, ResourceFunction["RamanujanC"][q, n] == \!\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(q\)]\(Boole[ CoprimeQ[j, q]] \*SuperscriptBox[\(E\), FractionBox[\(2 \[Pi]\ I\ n\ j\), \(q\)]]\)\) // FullSimplify]
Out[10]=

RamanujanC can be expressed as a Dirichlet convolution:

In[11]:=
With[{q = 14, n = 8}, ResourceFunction["RamanujanC"][q, n] == DirichletConvolve[K Boole[Divisible[n, K]], MoebiusMu[K], K, q]]
Out[11]=

Neat Examples (1) 

Visualize Ramanujan's sum over integer values:

In[12]:=
MatrixPlot[Array[ResourceFunction["RamanujanC"], {100, 100}]]
Out[12]=

Version History

  • 1.0.0 – 08 March 2021

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