--- title: "PrimePi" language: "en" type: "Symbol" summary: "PrimePi[x] gives the number of primes PrimePi[x] less than or equal to x." keywords: - distribution of primes - prime counting function - prime count - Deleglise-Rivat - Eratosthenes - Lagarias-Miller-Odlyzko - Lehmer - Legendre - Meissel - sieve - pi canonical_url: "https://reference.wolfram.com/language/ref/PrimePi.html" source: "Wolfram Language Documentation" related_guides: - title: "Prime Numbers" link: "https://reference.wolfram.com/language/guide/PrimeNumbers.en.md" - title: "Analytic Number Theory" link: "https://reference.wolfram.com/language/guide/AnalyticNumberTheory.en.md" - title: "Multiplicative Number Theory" link: "https://reference.wolfram.com/language/guide/MultiplicativeNumberTheory.en.md" - title: "Mathematical Functions" link: "https://reference.wolfram.com/language/guide/MathematicalFunctions.en.md" - title: "Number Theory" link: "https://reference.wolfram.com/language/guide/NumberTheory.en.md" - title: "Number Theoretic Functions" link: "https://reference.wolfram.com/language/guide/NumberTheoreticFunctions.en.md" related_functions: - title: "Prime" link: "https://reference.wolfram.com/language/ref/Prime.en.md" - title: "NextPrime" link: "https://reference.wolfram.com/language/ref/NextPrime.en.md" - title: "RiemannR" link: "https://reference.wolfram.com/language/ref/RiemannR.en.md" - title: "LogIntegral" link: "https://reference.wolfram.com/language/ref/LogIntegral.en.md" - title: "Zeta" link: "https://reference.wolfram.com/language/ref/Zeta.en.md" - title: "ZetaZero" link: "https://reference.wolfram.com/language/ref/ZetaZero.en.md" - title: "EulerPhi" link: "https://reference.wolfram.com/language/ref/EulerPhi.en.md" related_tutorials: - title: "Integer and Number Theoretic Functions" link: "https://reference.wolfram.com/language/tutorial/MathematicalFunctions.en.md#20170" - title: "Implementation Notes: Numerical and Related Functions" link: "https://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.en.md#12788" --- # PrimePi PrimePi[x] gives the number of primes $\pi (x)$ less than or equal to x. ## Details and Options * ``PrimePi`` is also known as prime counting function. * Mathematical function, suitable for both symbolic and numerical manipulation. * $\pi (x)$ counts the prime numbers less than or equal to ``x``. * $\pi (x)$ has the asymptotic expansion $x/\log (x)+x\left/\log ^2(x)\right.+2 x\left/\log ^3(x)\right.+\ldots$ as $x\to \infty$. [image] * The following option can be given: | | | | | ------------------ | ------------------- | ------------------------------------------------- | | Method | Automatic | method to use | | ProgressReporting | \$ProgressReporting | whether to report the progress of the computation | * Possible settings for ``Method`` include: | | | | ---------------- | ----------------------------------------- | | "DelegliseRivat" | use the Deléglise–Rivat algorithm | | "Legendre" | use the Legendre formula | | "Lehmer" | use the Lehmer formula | | "LMO" | use the Lagarias–Miller–Odlyzko algorithm | | "Meissel" | use the Meissel formula | | "Sieve" | use the sieve of Erastosthenes | --- ## Examples (50) ### Basic Examples (3) Compute the number of primes up to 15: ```wl In[1]:= PrimePi[15] Out[1]= 6 ``` --- Plot the prime counting function: ```wl In[1]:= Plot[PrimePi[x], {x, 0, 50}] Out[1]= [image] ``` --- Find the leading asymptotic term for ``PrimePi`` at ``Infinity`` : ```wl In[1]:= Asymptotic[PrimePi[x], x -> Infinity] Out[1]= (x/Log[x]) ``` ### Scope (10) #### Numerical Evaluation (5) ``PrimePi`` works over integers: ```wl In[1]:= PrimePi[100] Out[1]= 25 ``` --- Rational numbers: ```wl In[1]:= PrimePi[7 / 2] Out[1]= 2 ``` --- Real numbers: ```wl In[1]:= PrimePi[15.25] Out[1]= 6 ``` --- ``PrimePi`` works over large numbers: ```wl In[1]:= PrimePi[10 ^ 15] Out[1]= 29844570422669 ``` --- ``PrimePi`` threads over lists: ```wl In[1]:= PrimePi[{3, 5, 17, 25}] Out[1]= {2, 3, 7, 9} ``` #### Symbolic Manipulation (5) The traditional mathematical notation for ``PrimePi`` : ```wl In[1]:= PrimePi[x]//TraditionalForm Out[1]//TraditionalForm= $$\pi (x)$$ ``` --- Find a solution instance of equalities with ``PrimePi`` : ```wl In[1]:= FindInstance[PrimePi[n] == 7, n, Integers] Out[1]= {{n -> 17}} ``` --- Evaluate an integral: ```wl In[1]:= Integrate[PrimePi[x ^ 2], {x, 0, 10}] Out[1]= 250 - Sqrt[2] - Sqrt[3] - Sqrt[5] - Sqrt[7] - Sqrt[11] - Sqrt[13] - Sqrt[17] - Sqrt[19] - Sqrt[23] - Sqrt[29] - Sqrt[31] - Sqrt[37] - Sqrt[41] - Sqrt[43] - Sqrt[47] - Sqrt[53] - Sqrt[59] - Sqrt[61] - Sqrt[67] - Sqrt[71] - Sqrt[73] - Sqrt[79] - Sqrt[83] - Sqrt[89] - Sqrt[97] ``` --- Recognize the ``PrimePi`` function: ```wl In[1]:= FindSequenceFunction[{0, 1, 2, 2, 3, 3, 4, 4, 4, 4}, n] Out[1]= PrimePi[n] ``` --- Find asymptotic relations: ```wl In[1]:= Asymptotic[PrimePi[x], x -> Infinity] Out[1]= (x/Log[x]) ``` Verify asymptotic relations: ```wl In[2]:= AsymptoticEquivalent[PrimePi[x], LogIntegral[x], x -> Infinity] Out[2]= True ``` ### Options (6) #### Method (5) Specify the Legendre method to be used for counting primes [[MathWorld](http://mathworld.wolfram.com/LegendresFormula.html)]: ```wl In[1]:= PrimePi[10 ^ 6, Method -> "Legendre"] Out[1]= 78498 ``` Compare the timing: ```wl In[2]:= BarChart[Table[{First[AbsoluteTiming[PrimePi[10 ^ n]]], First[AbsoluteTiming[PrimePi[10 ^ n, Method -> "Legendre"]]]}, {n, 7, 11, .5}], ChartLegends -> {"Automatic", "Legendre"}] Out[2]= [image] ``` --- Specify the Lehmer method to be used for counting primes [[MathWorld](http://mathworld.wolfram.com/LehmersFormula.html)]: ```wl In[1]:= PrimePi[10 ^ 6, Method -> "Lehmer"] Out[1]= 78498 ``` Compare the timing: ```wl In[2]:= BarChart[Table[{First[AbsoluteTiming[PrimePi[10 ^ n]]], First[AbsoluteTiming[PrimePi[10 ^ n, Method -> "Lehmer"]]]}, {n, 7, 11, .5}], ChartLegends -> {"Automatic", "Lehmer"}] Out[2]= [image] ``` --- Specify the Deléglise–Rivat method to be used for counting primes: ```wl In[1]:= PrimePi[10 ^ 6, Method -> "DelegliseRivat"] Out[1]= 78498 ``` Compare the timing: ```wl In[2]:= BarChart[Table[{First[AbsoluteTiming[PrimePi[10 ^ n]]], First[AbsoluteTiming[PrimePi[10 ^ n, Method -> "DelegliseRivat"]]]}, {n, 7, 11, .5}], ChartLegends -> {"Automatic", "Deleglise-Rivat"}] Out[2]= [image] ``` --- Specify the Meissel method to be used for counting primes [[MathWorld](http://mathworld.wolfram.com/MeisselsFormula.html)]: ```wl In[1]:= PrimePi[10 ^ 6, Method -> "Meissel"] Out[1]= 78498 ``` Compare the timing: ```wl In[2]:= BarChart[Table[{First[AbsoluteTiming[PrimePi[10 ^ n]]], First[AbsoluteTiming[PrimePi[10 ^ n, Method -> "Meissel"]]]}, {n, 7, 11, .5}], ChartLegends -> {"Automatic", "Meissel"}] Out[2]= [image] ``` --- Specify the Lagarias–Miller–Odlyzko (LMO) to be used for counting primes: ```wl In[1]:= PrimePi[10 ^ 6, Method -> "LMO"] Out[1]= 78498 ``` Compare the timing: ```wl In[2]:= BarChart[Table[{First[AbsoluteTiming[PrimePi[10 ^ n]]], First[AbsoluteTiming[PrimePi[10 ^ n, Method -> "LMO"]]]}, {n, 7, 11, .5}], ChartLegends -> {"Automatic", "Lagarias-Miller-Odlyzko"}] Out[2]= [image] ``` #### ProgressReporting (1) By default, ``PrimePi`` does not report progress: ```wl In[1]:= PrimePi[10 ^ 17 + 1] Out[1]= 2623557157654233 ``` With the setting ``ProgressReporting -> True``, ``PrimePi`` shows the progress of the computation: ```wl In[2]:= PrimePi[10 ^ 17 + 1, ProgressReporting -> True] During evaluation of In[7]:= [image] ``` ### Applications (22) #### Basic Applications (7) Visualize the ``PrimePi`` function: ```wl In[1]:= Grid[Prepend[Table[{"10" ^ n, PrimePi[10 ^ n]}, {n, 1, 10}], {"x", "π(x)"}], ...] Out[1]= | | | | ------- | --------- | | "x" | "π(x)" | | "10" | 4 | | "10"^2 | 25 | | "10"^3 | 168 | | "10"^4 | 1229 | | "10"^5 | 9592 | | "10"^6 | 78498 | | "10"^7 | 664579 | | "10"^8 | 5761455 | | "10"^9 | 50847534 | | "10"^10 | 455052511 | ``` --- Spiral of ``PrimePi`` : ```wl In[1]:= ListPolarPlot[Table[{Mod[π / 2 - n 4π / 50, 2π], PrimePi[n]}, {n, 50}], ...] Out[1]= [image] ``` --- Hexagonal prime spiral: ```wl In[1]:= hexagonalSpiral[n_] := {Re[#], Im[#]}& /@ Fold[Join[#1, Last[#1] + Exp[I Pi / 3] ^ #2Range[#2 / 2]]&, {0}, Range[n]] In[2]:= spiral = hexagonalSpiral[25]; In[3]:= primes = spiral[[Prime[Range[PrimePi[Length[spiral]]]]]]; In[4]:= ListPlot[spiral, ...] Out[4]= [image] ``` --- ``ContourPlot`` of the ``PrimePi`` function: ```wl In[1]:= ListContourPlot[Table[{x = n, y = PrimePi[n]}, {n, 1, 25}]] Out[1]= [image] ``` --- Plot the difference between ``RiemannR`` and the ``PrimePi`` function: ```wl In[1]:= r = 0;c = -1;ListPolarPlot[{Table[{i * Pi / 8, x = RiemannR[i] - PrimePi[i];If[x > c, c = x;r = r + 1];PrimePi[i] * r / i}, {i, 1000}]}, Rule[...]] Out[1]= [image] ``` --- Find all prime numbers less than or equal to 100: ```wl In[1]:= Prime[Range[PrimePi[100]]] Out[1]= {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} ``` --- Count the number of primes in an interval: ```wl In[1]:= primecount[{x_, y_}] := PrimePi[y + 1] - PrimePi[x - 1] In[2]:= primecount[{5, 15}] Out[2]= 4 ``` #### Approximations (7) Approximations to ``PrimePi`` : ```wl In[1]:= PrimePi[5000] Out[1]= 669 In[2]:= 5000 / Log[5000.] Out[2]= 587.048 In[3]:= LogIntegral[5000.] Out[3]= 684.281 In[4]:= RiemannR[5000.] Out[4]= 668.946 ``` Plot ``PrimePi`` compared with estimates: ```wl In[5]:= Plot[{PrimePi[x], x / Log[x], LogIntegral[x], RiemannR[x]}, {x, 1.5, 100}, Rule[...]] Out[5]= [image] ``` --- $\pi (x)$ is bounded below by $\frac{x}{\log (x)}$ and is bounded above by $\frac{x}{\log (x)-1.112}$ for $x$ greater than 3: ```wl In[1]:= Plot[{x / Log[x], PrimePi[x], x / (Log[x] - 1.112)}, {x, 3, 100}, Rule[...]] Out[1]= [image] ``` --- $\frac{ x}{H_x-3/2}$ is within $\approx 2$ of $\pi (x)$ for $50\leq x\leq 1000$ : ```wl In[1]:= Plot[{(x/HarmonicNumber[x] - 3 / 2), PrimePi[x]}, {x, 50, 200}, Rule[...]] Out[1]= [image] ``` --- $| \pi (x)-\text{li}(x)| <\frac{\sqrt{x} \log (x)}{8 \pi }$ for $x>2657$ if the Riemann hypothesis is true: ```wl In[1]:= Plot[{Abs[PrimePi[x] - LogIntegral[x]], Sqrt[x]Log[x] / (8 * Pi)}, {x, 2657, 3000}, PlotLegends -> "Expressions"] Out[1]= [image] ``` --- Count the prime numbers using ``EulerPhi`` : ```wl In[1]:= primecount[x_] := Underoverscript[∑, k = 2, ⌊x⌋]⌊(EulerPhi[k]/k - 1)⌋ In[2]:= Table[primecount[n], {n, 0, 10}] Out[2]= {0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4} In[3]:= Table[PrimePi[n], {n, 0, 10}] Out[3]= {0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4} ``` --- Compute ``PrimePi`` based on the Hardy–Wright formula: ```wl In[1]:= primecount[x_] := Sum[Mod[(j - 2)!, j], {j, 4, x}] In[2]:= Table[primecount[n], {n, 5, 10}] Out[2]= {3, 3, 4, 4, 4, 4} In[3]:= Table[PrimePi[n], {n, 5, 10}] Out[3]= {3, 3, 4, 4, 4, 4} ``` --- Compute ``PrimePi`` using ``Accumulate`` : ```wl In[1]:= Accumulate[Table[Boole[PrimeQ[n]], {n, 1, 10}]] Out[1]= {0, 1, 2, 2, 3, 3, 4, 4, 4, 4} In[2]:= Table[PrimePi[n], {n, 1, 10}] Out[2]= {0, 1, 2, 2, 3, 3, 4, 4, 4, 4} ``` #### Number Theory (8) Find twin primes up to $n$, i.e. pairs of primes of the form $(p,p+2)$ : ```wl In[1]:= TwinPrimes[n_] := Pick[#, PrimeQ[# + 2]]&[Prime[Range[PrimePi[n]]]] In[2]:= TwinPrimes[100] Out[2]= {3, 5, 11, 17, 29, 41, 59, 71} In[3]:= TwinPrimePi[n_] := Length[Pick[#, PrimeQ[# + 2]]]&[Prime[Range[PrimePi[n]]]] ``` Plot the sequence of twin primes and ``PrimePi`` : ```wl In[4]:= ListStepPlot[{Table[TwinPrimePi[n], {n, 1, 100}], Table[PrimePi[n], {n, 1, 100}]}] Out[4]= [image] ``` --- The $n$$$^{\text{th}}$$ Ramanujan prime is the smallest number $R_n$ such that $\pi x-\frac{\pi x}{2}\geq n$ for all $x\geq R_n$ : ```wl In[1]:= list = Table[PrimePi[n] - PrimePi[n / 2], {n, 10 ^ 4}]; In[2]:= ramanujanPrime = 1 + Last[Position[list, #]][[1]]& /@ Range[0, 50]; In[3]:= ramanujanPrimePi[n_] := Count[ramanujanPrime, _ ? (# ≤ n&)] ``` Plot the sequence of Ramanujan primes: ```wl In[4]:= Plot[ramanujanPrimePi[n], {n, 0, 100}] Out[4]= [image] ``` Compare the count of Ramanujan primes to ``PrimePi`` : ```wl In[5]:= Plot[{ramanujanPrimePi[n], PrimePi[n]}, {n, 0, 100}] Out[5]= [image] ``` --- Calculate the primorial up to the $n$$$^{\text{th}}$$ prime, i.e. a function that multiplies successive primes, similar to the factorial: ```wl In[1]:= primorial[0] := 1 primorial[1] := 2 primorial[n_] := Times@@Prime@Range[n]; In[2]:= primorial[5] Out[2]= 2310 ``` Compare the primorial to the factorial up to $n$ : ```wl In[3]:= DiscretePlot[{Log[n!], Log[Times@@Prime[Range[PrimePi[n]]]]}, {n, 1, 50}] Out[3]= [image] ``` Plot the differences between the factorial and the primorial up to $n$* : * ```wl In[4]:= ListLinePlot[If[True, Differences, Identity][Table[Log[(n!/Times@@Prime[Range[PrimePi[n]]])], {n, 1, 100, 1}]], Filling -> Axis] Out[4]= [image] ``` Plot the Chebyshev theta function: ```wl In[5]:= chebyshevTheta[n_] := Log[primorial[PrimePi[n]]]; In[6]:= Plot[chebyshevTheta[n], {n, 1, 100}] Out[6]= [image] In[7]:= DiscretePlot[{chebyshevTheta[n], PrimePi[n]}, {n, 1, 100}] Out[7]= [image] ``` --- Calculate the prime powers up to $n$ : ```wl In[1]:= primePowers[n_] := Union@@Table[Prime[Range[PrimePi[n ^ (1 / x)]]] ^ x, {x, Log2[n]}] In[2]:= primePowers[10] Out[2]= {2, 3, 4, 5, 7, 8, 9} In[3]:= PrimePowerQ[%] Out[3]= {True, True, True, True, True, True, True} ``` Count all the prime powers up to $n$ : ```wl In[4]:= primePowerPi[n_] := PrimePi[n] + Sum[Floor@Log[Prime[k], n] - 1, {k, PrimePi[Sqrt[n]//Floor]}] In[5]:= primePowerPi[10] Out[5]= 7 ``` Graph the count of prime powers: ```wl In[6]:= DiscretePlot[primePowerPi[n], {n, 1, 100}] Out[6]= [image] ``` Compare the count of prime powers to ``PrimePi`` : ```wl In[7]:= DiscretePlot[{primePowerPi[n], PrimePi[n]}, {n, 1, 100}] Out[7]= [image] ``` --- Visualize the second Hardy–Littlewood conjecture, which states that $\pi (x+y)\leq \pi (x)+\pi (y)$ for $x,y\geq 2$ : ```wl In[1]:= ArrayPlot[((Outer[Plus, #, #]&)[PrimePi[Range[2, 20]]] - Outer[PrimePi[+##1]&, #, #]&)[Range[2, 20]], Rule[...]] Out[1]= [image] ``` --- Plot Brocard's conjecture, which states that if $p_n$ and $p_{n+1}$ are consecutive primes greater than 2, then between $p_n^2$ and $p_{n+1}^2$ there are at least four prime numbers: ```wl In[1]:= ListStepPlot[-Subtract@@@Partition[PrimePi[Prime[Range[100]] ^ 2], 2, 1], Rule[...]] Out[1]= [image] ``` --- Find Goldbach partitions, i.e. pairs of primes ($p$, $q$) such that $2n=p+q$ : ```wl In[1]:= goldbachPartitions[n_] := IntegerPartitions[2n, {2}, Prime[Range[PrimePi[2n]]]] In[2]:= goldbachPartitions[50] Out[2]= {{97, 3}, {89, 11}, {83, 17}, {71, 29}, {59, 41}, {53, 47}} ``` Graph Goldbach's conjecture/comet: ```wl In[3]:= ListPlot[Transpose[{Range[4, 1000, 2], Length /@ Table[IntegerPartitions[i, 2, Table[Prime[n], {n, PrimePi[i]}]], {i, 4, 1000, 2}]}], Rule[...]] Out[3]= [image] ``` --- The only 8 solutions of $\pi (x)=\phi (x)$ : ```wl In[1]:= Cases[Range[100], n_ /; PrimePi[n] == EulerPhi[n]] Out[1]= {2, 3, 4, 8, 10, 14, 20, 90} In[2]:= DiscretePlot[{PrimePi[n], EulerPhi[n]}, {n, 1, 22}, Rule[...]] Out[2]= [image] ``` ### Properties & Relations (5) The largest domain of definitions of ``PrimePi``: ```wl In[1]:= FunctionDomain[PrimePi[x], x, Complexes] Out[1]= x∈ℝ ``` --- ``PrimePi`` is asymptotically equivalent to $\frac{x}{\log (x)}$ as $x\to \infty$ : ```wl In[1]:= AsymptoticEquivalent[PrimePi[x], x / Log[x], x -> Infinity] Out[1]= True ``` And to ``LogIntegral`` : ```wl In[2]:= AsymptoticEquivalent[PrimePi[x], LogIntegral[x], x -> Infinity] Out[2]= True ``` --- ``PrimePi`` is the inverse of ``Prime`` : ```wl In[1]:= PrimePi[Prime[100]] Out[1]= 100 ``` --- Use ``PrimeQ`` to count prime numbers: ```wl In[1]:= Count[Table[PrimeQ[n], {n, 1, 1000}], True] Out[1]= 168 In[2]:= PrimePi[1000] Out[2]= 168 ``` --- Mathematical function entity: ```wl In[1]:= Entity["MathematicalFunction", "PrimePi"] Out[1]= Entity["MathematicalFunction", "PrimePi"] ``` An integral representation of ``PrimePi`` : ```wl In[2]:= Entity["MathematicalFunction", "PrimePi"]["IntegralRepresentations"] Out[2]= {Function[{\[FormalN]}, Inactivate[ConditionalExpression[PrimePi[\[FormalN]] == \[FormalN] - 1 - (Subsuperscript[∫, 0, 2 π](Underoverscript[∑, \[FormalM] = 1, \[FormalN]]Cos[\[FormalT] Underoverscript[∏, \[FormalK] = 1, \[FormalM] - 1]Underoverscript[∏, \[FormalJ] = 1, \[FormalM] - 1](\[FormalJ] \[FormalK] - \[FormalM])])\[DifferentialD]\[FormalT]/2 π), \[FormalN]∈ℤ && \[FormalN] > 0]]]} ``` ### Possible Issues (1) Evaluation time increases exponentially: ```wl In[1]:= ListLinePlot[Table[First[AbsoluteTiming[PrimePi[10 ^ n]]], {n, 14, 19, .5}], ...] Out[1]= [image] ``` ### Neat Examples (3) Ulam spiral colored based on the difference in ``PrimePi`` values: ```wl In[1]:= ulam[n_] := Partition[Permute[Range[n ^ 2], Accumulate[Take[Flatten[{{n ^ 2 + 1} / 2, Table [(-1) ^ j i, {j, n}, {i, {-1, n}}, {j}]}], n ^ 2]]], n] In[2]:= ArrayPlot[PrimePi[ulam[101] + 10] - PrimePi[ulam[101]], Rule[...]] Out[2]= [image] ``` --- Generate a path based on the ``PrimePi`` function: ```wl In[1]:= Graphics[Line[AnglePath[Table[PrimePi[n], {n, 1, 1000}]], Rule[...]]] Out[1]= [image] ``` --- Construct polyhedra using directed graphs generated by primes less than $p$ : ```wl In[1]:= vertexcoords[p_] := GraphEmbedding[Graph3D[Flatten[Table[i -> Mod[i * q, p], {i, 1, p - 1}, {q, Prime[Range[PrimePi[p] - 1]]}]]]] In[2]:= Table[ConvexHullMesh[vertexcoords[p], Rule[...]], {p, Prime[Range[4, 7]]}] Out[2]= {[image], [image], [image], [image]} ``` ## See Also * [`Prime`](https://reference.wolfram.com/language/ref/Prime.en.md) * [`NextPrime`](https://reference.wolfram.com/language/ref/NextPrime.en.md) * [`RiemannR`](https://reference.wolfram.com/language/ref/RiemannR.en.md) * [`LogIntegral`](https://reference.wolfram.com/language/ref/LogIntegral.en.md) * [`Zeta`](https://reference.wolfram.com/language/ref/Zeta.en.md) * [`ZetaZero`](https://reference.wolfram.com/language/ref/ZetaZero.en.md) * [`EulerPhi`](https://reference.wolfram.com/language/ref/EulerPhi.en.md) ## Tech Notes * [Integer and Number Theoretic Functions](https://reference.wolfram.com/language/tutorial/MathematicalFunctions.en.md#20170) * [Implementation Notes: Numerical and Related Functions](https://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.en.md#12788) ## Related Guides * [Prime Numbers](https://reference.wolfram.com/language/guide/PrimeNumbers.en.md) * [Analytic Number Theory](https://reference.wolfram.com/language/guide/AnalyticNumberTheory.en.md) * [Multiplicative Number Theory](https://reference.wolfram.com/language/guide/MultiplicativeNumberTheory.en.md) * [Mathematical Functions](https://reference.wolfram.com/language/guide/MathematicalFunctions.en.md) * [Number Theory](https://reference.wolfram.com/language/guide/NumberTheory.en.md) * [Number Theoretic Functions](https://reference.wolfram.com/language/guide/NumberTheoreticFunctions.en.md) ## Related Links * [MathWorld](http://mathworld.wolfram.com/PrimeCountingFunction.html) * [An Elementary Introduction to the Wolfram Language: More about Numbers](https://www.wolfram.com/language/elementary-introduction/23-more-about-numbers.html) * [NKS\|Online](http://www.wolframscience.com/nks/search/?q=PrimePi) * [A New Kind of Science](http://www.wolframscience.com/nks/) ## History * Introduced in 1991 (2.0) \| [Updated in 2020 (12.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn121.en.md) ▪ [2021 (12.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn123.en.md)