# Meissel–Mertens constant

Meissel–Mertens constant
You are encouraged to solve this task according to the task description, using any language you may know.

Calculate Meissel–Mertens constant up to a precision your language can handle.

Motivation

Analogous to Euler's constant, which is important in determining the sum of reciprocal natural numbers, Meissel-Mertens' constant is important in calculating the sum of reciprocal primes.

Example

We consider the finite sum of reciprocal natural numbers:

1 + 1/2 + 1/3 + 1/4 + 1/5 ... 1/n

this sum can be well approximated with:

log(n) + E

where E denotes Euler's constant: 0.57721...

log(n) denotes the natural logarithm of n.

Now consider the finite sum of reciprocal primes:

1/2 + 1/3 + 1/5 + 1/7 + 1/11 ... 1/p

this sum can be well approximated with:

log( log(p) ) + M

where M denotes Meissel-Mertens constant: 0.26149...

See

## 11l

Translation of: Python
```F primes_up_to_limit(Int limit)
[Int] r
I limit >= 2
r.append(2)

V isprime = [1B] * ((limit - 1) I/ 2)
V sieveend = Int(sqrt(limit))
L(i) 0 .< isprime.len
I isprime[i]
Int p = i * 2 + 3
r.append(p)
I i <= sieveend
L(j) ((p * p - 3) >> 1 .< isprime.len).step(p)
isprime[j] = 0B
R r

V euler = 0.57721566490153286
V m = 0.0
L(x) primes_up_to_limit(10'000'000)
m += log(1 - (1 / x)) + (1 / x)

print(‘MM = #.16’.format(euler + m))```
Output:
```MM = 0.2614972157776471
```

## ALGOL 68

Sieving odd primes only, not using extended precision.
Note that to run this with Algol 68G, a large heap size must be specified, e.g.: by adding `-heap 256M` to the command line.

```BEGIN # compute an approximation to the Meissel-Mertens constant             #
# construct a sieve of odd primes                                        #
[ 0 : 10 000 000 ]BOOL primes;
BEGIN
FOR i TO UPB primes DO primes[ i ] := TRUE  OD;
INT ip := 1;
FOR i WHILE i + ( ip +:= 2 ) <= UPB primes DO
IF primes[ i ] THEN
FOR s FROM i + ip BY ip TO UPB primes DO primes[ s ] := FALSE OD
FI
OD
END;
# sum the reciprocals of the primes                                      #
INT       p count := 1;
INT       last p  := 0;
LONG REAL sum     := long ln( 0.5 ) + 0.5;
INT       p       := 1;
INT       p10     := 10;
# Euler's constant from the wikipedia, truncated for LONG REAL           #
LONG REAL eulers constant = 0.5772156649015328606 # 0651209008240243104215933593992 #;
FOR i TO UPB primes DO
p +:= 2;
IF primes[ i ] THEN
LONG REAL rp = 1 / LENG p;
sum     +:= long ln( 1 - rp ) + rp;
p count +:= 1;
last p   := p;
IF p count = p10 THEN
print( ( "after ", whole( p count, -8 ), " primes, the approximation is: "
, fixed( sum + eulers constant, -14, 12 )
, ", last prime considered: ", whole( last p, 0 )
, newline
)
);
p10 := IF p10 < 1 000 000 THEN p10 * 10 ELSE p10 + 1 000 000 FI
FI
FI
OD;
print( ( "after ", whole( p count, -8 ), " primes, the approximation is: "
, fixed( sum + eulers constant, -14, 12 )
, ", last prime considered: ", whole( last p, 0 )
, newline
)
)
END```
Output:
```after       10 primes, the approximation is: 0.265160104017, last prime considered: 29
after      100 primes, the approximation is: 0.261624626173, last prime considered: 541
after     1000 primes, the approximation is: 0.261503563617, last prime considered: 7919
after    10000 primes, the approximation is: 0.261497594498, last prime considered: 104729
after   100000 primes, the approximation is: 0.261497238454, last prime considered: 1299709
after  1000000 primes, the approximation is: 0.261497214692, last prime considered: 15485863
after  1270607 primes, the approximation is: 0.261497214255, last prime considered: 19999999
```

## ALGOL W

Translation of: ALGOL 68
```begin % compute an approximation to the Meissel-Mertens constant             %
integer MAX_PRIME;
MAX_PRIME := 10000000;
begin
logical array primes ( 0 :: MAX_PRIME );
begin % construct a sieve of odd primes                                  %
integer i, ip;
for i := 1 until MAX_PRIME do primes( i ) := true;
ip := 3;
i  := 1;
while i + ip <= MAX_PRIME do begin
if primes( i ) then begin
for s := i + ip step ip until MAX_PRIME do primes( s ) := false
end if_primes__i ;
ip := ip + 2;
i  := i + 1
end while_i_plus_ip_le_MAX_PRIME
end;
begin % sum the reciprocals of the primes                                %
integer   pCount, lastP, p, p10;
long real sum, eulersConstant;
procedure showProgress ;
write( s_w := 0, i_w := 8
, r_format := "A", r_w := 14, r_d := 12
, "after ", pCount, " primes, the approximation is: "
, ( sum + eulersConstant ), ", last prime considered: "
, i_w := 1
, lastP
);
pCount := 1;
lastP  := 0;
sum    := longLn( 0.5 ) + 0.5;
p      := 1;
p10    := 10;
% Euler's constant from the wikipedia, truncated for long real       %
eulersConstant := 0.5772156649015328606 % 0651209008240243104215933593992 %;
for i := 1 until MAX_PRIME do begin
p := p + 2;
if primes( i ) then begin
long real rp;
rp      := 1 / long p;
sum     := sum + longLn( 1 - rp ) + rp;
pCount  := pCount + 1;
lastP   := p;
if pCount = p10 then begin
showProgress;
p10 := if p10 < 1000000 THEN p10 * 10 else p10 + 1000000
end if_pCount_eq_p10
end if_primes__i
end for_i ;
showProgress
end
end
end.```
Output:
```after       10 primes, the approximation is: 0.265160104017, last prime considered: 29
after      100 primes, the approximation is: 0.261624626173, last prime considered: 541
after     1000 primes, the approximation is: 0.261503563617, last prime considered: 7919
after    10000 primes, the approximation is: 0.261497594498, last prime considered: 104729
after   100000 primes, the approximation is: 0.261497238454, last prime considered: 1299709
after  1000000 primes, the approximation is: 0.261497214692, last prime considered: 15485863
after  1270607 primes, the approximation is: 0.261497214255, last prime considered: 19999999
```

## Arturo

```meisselMertens: function [depth][
Euler: 0.57721566490153286
m: (1//2) + ln 1-1//2
loop range.step:2 3 depth 'x ->
if prime? x ->
m: m + (1//x) + ln 1-1//x

return m + Euler
]

print meisselMertens 10000000
```
Output:
`0.2614972157776471`

## BASIC

### BASIC256

Translation of: FreeBASIC
```Euler = 0.5772156649
m = 0
for x = 2 to 1e6    # more prime numbers do not add more precision
if isPrime(x) then m += log(1-(1/x)) + (1/x)
next x
print "MM = "; Euler + m
print Euler
end

function isPrime(v)
if v < 2 then return False
if v mod 2 = 0 then return v = 2
if v mod 3 = 0 then return v = 3
d = 5
while d * d <= v
if v mod d = 0 then return False else d += 2
end while
return True
end function
```
Output:
`MM = 0.26149724673`

### Run BASIC

Translation of: FreeBASIC
```function isPrime(n)
if n < 2       then isPrime = 0 : goto [exit]
if n = 2       then isPrime = 1 : goto [exit]
if n mod 2 = 0 then isPrime = 0 : goto [exit]
isPrime = 1
for i = 3 to int(n^.5) step 2
if n mod i = 0 then isPrime = 0 : goto [exit]
next i
[exit]
end function

e = 0.5772156

for x = 2 to 100000    ' more prime numbers do not add more precision
if isPrime(x) then m = m + log(1-(1/x)) + (1/x)
next x
print "MM = "; e + m```
Output:
`MM = 0.261274`

### PureBasic

Translation of: FreeBASIC
```Procedure isPrime(v.i)
If     v <= 1    : ProcedureReturn #False
ElseIf v < 4     : ProcedureReturn #True
ElseIf v % 2 = 0 : ProcedureReturn #False
ElseIf v < 9     : ProcedureReturn #True
ElseIf v % 3 = 0 : ProcedureReturn #False
Else
Protected r = Round(Sqr(v), #PB_Round_Down)
Protected f = 5
While f <= r
If v % f = 0 Or v % (f + 2) = 0
ProcedureReturn #False
EndIf
f + 6
Wend
EndIf
ProcedureReturn #True
EndProcedure

OpenConsole()
Euler.d = 0.5772156649 ;0153286

For x.i = 2 To 1e8
If isPrime(x)
m.d = m + Log(1-(1/x)) + (1/x)
EndIf
Next x
PrintN("MM = " + StrD(Euler + m))
PrintN(#CRLF\$ + "--- terminado, pulsa RETURN---"): Input()
CloseConsole()
```
Output:
`MM = 0.2614972129`

### True BASIC

Translation of: FreeBASIC
```FUNCTION isPrime (n)
IF n = 2 THEN
LET isPrime = 1
ELSEIF n <= 1 OR REMAINDER(n, 2) = 0 THEN
LET isPrime = 0
ELSE
LET isPrime = 1
FOR i = 3 TO INT(SQR(n)) STEP 2
IF REMAINDER(n, i) = 0 THEN
LET isPrime = 0
EXIT FUNCTION
END IF
NEXT i
END IF
END FUNCTION

LET e = .5772156649

FOR x = 2 to 1e6                  !more prime numbers do not add more precision
IF isPrime(x) = 1 THEN
LET m = m + LOG(1-(1/x)) + (1/x)
END IF
NEXT x
PRINT "MM ="; e + m
END
```
Output:
`MM = 0.26149725`

### Yabasic

Translation of: FreeBASIC
```e = 0.5772156

for x = 2 to 1e6    // more prime numbers do not add more precision
if isPrime(x)  m = m + log(1-(1/x)) + (1/x)
next x
print "MM = ", e + m
end

sub isPrime(v)
if v < 2  return False
if mod(v, 2) = 0  return v = 2
if mod(v, 3) = 0  return v = 3
d = 5
while d * d <= v
if mod(v, d) = 0 then return False else d = d + 2 : fi
wend
return True
end sub
```
Output:
`MM = 0.261497`

## C++

```#include <cmath>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <vector>

std::vector<double> list_prime_reciprocals(const int32_t& limit) {
const int32_t half_limit = ( limit % 2 == 0 ) ? limit / 2 : 1 + limit / 2;
std::vector<bool> composite(half_limit);
for ( int32_t i = 1, p = 3; i < half_limit; p += 2, ++i ) {
if ( ! composite[i] ) {
for ( int32_t a = i + p; a < half_limit; a = a + p ) {
composite[a] = true;
}
}
}

std::vector<double> result(composite.size());
result[0] = 0.5;
for ( int32_t i = 1, p = 3; i < half_limit; p += 2, ++i ) {
if ( ! composite[i] ) {
result.emplace_back(1.0 / p);
}
}
return result;
}

int main() {
std::vector<double> prime_reciprocals = list_prime_reciprocals(100000000);
const double euler = 0.577215664901532861;
double sum = 0.0;
for ( double reciprocal : prime_reciprocals ) {
sum += reciprocal + log(1.0 - reciprocal);
}

const double meissel_mertens = euler + sum;
std::cout << "The Meissel-Mertens constant = " << std::setprecision(8) << meissel_mertens << std::endl;
}
```
Output:
```
The Meissel-Mertens constant = 0.26149721
```

## Delphi

Works with: Delphi version 6.0

This program doesn't test the limits of numbers in Delphi. The limit is the processing time. The time to process 10^7 is 3 minutes. The time to process 10^8 is nearly two hours. As a result, process beyond 10^8 is pretty much impossible and a different strategy would be required

```function IsPrime(N: int64): boolean;
{Fast, optimised prime test}
var I,Stop: int64;
begin
if (N = 2) or (N=3) then Result:=true
else if (n <= 1) or ((n mod 2) = 0) or ((n mod 3) = 0) then Result:= false
else
begin
I:=5;
Stop:=Trunc(sqrt(N+0.0));
Result:=False;
while I<=Stop do
begin
if ((N mod I) = 0) or ((N mod (I + 2)) = 0) then exit;
Inc(I,6);
end;
Result:=True;
end;
end;

function GetNextPrime(var Start: integer): integer;
{Get the next prime number after Start}
{Start is passed by "reference," so the
{original variable is incremented}
begin
repeat Inc(Start)
until IsPrime(Start);
Result:=Start;
end;

function MeisselMertens(Depth: integer; Prog: TProgress): extended;
{Calculate MM value up a certain Depth}
var I,P: integer;
const Euler = 0.57721566490153286;
begin
Result:=0;
P:=1;
for I:=1 to Depth do
begin
P:=GetNextPrime(P);
Result:=Result+Ln(1-(1/P)) + (1/P);
if Assigned(Prog) and ((I mod 10000)=0) then
HandleProgress(MulDiv(100,I,Depth));
end;
Result:=Result+Euler;
end;

procedure ShowMeisselMertens(Memo: TMemo; Prog: TProgress);
var I,IT,Digits: integer;
var M,Last: extended;
begin
Last:=0;
IT:=1;
{Calculate MM to specified Power of 10}
for I:=1 to 7 do
begin
IT:=IT*10;
M:=MeisselMertens(IT,Prog);
{Calculated Digits of accuracy}
Digits:=Trunc(abs(Log(abs(M-Last))));
Last:=M
end;
end;
```
Output:
```Primes  Digits       M
-----------------------------------------
10^ 1        0       0.265160104017311294
10^ 2        2       0.261624626172936147
10^ 3        3       0.261503563616850571
10^ 4        5       0.261497594498432488
10^ 5        6       0.261497238454297260
10^ 6        7       0.261497214692305277
10^ 7        8       0.261497212987251183
10^ 8        9       0.261497212858582276

```

## Dart

Translation of: FreeBASIC
```import 'dart:math';

bool isPrime(var n) {
if (n <= 1) return false;
if (n == 2) return true;
for (var i = 2; i <= sqrt(n); i++) if (n % i == 0) return false;
return true;
}

void main() {
const double euler = 0.57721566490153286;

double m = 0.0;
for (var x = 2; x <= 1e8; x++)
if (isPrime(x)) m += log(1 - (1 / x)) + (1 / x);

print('MM = \${euler + m}');
}
```
Output:
`MM = 0.2614972131057144`

## EasyLang

Translation of: BASIC256
```fastfunc isprim num .
if num mod 2 = 0
if num = 2
return 1
.
return 0
.
i = 3
while i <= sqrt num
if num mod i = 0
return 0
.
i += 2
.
return 1
.
func log x .
return log10 x / log10 2.7182818284590452354
.
euler = 0.5772156649
for x = 2 to 1e6
if isprim x = 1
m += log (1 - (1 / x)) + (1 / x)
.
.
numfmt 11 0
print "mm = " & euler + m
```
Output:
```mm = 0.26149724673
```

## FreeBASIC

On my CPU i5 it takes about 3.5 minutes

```'#include "isprime.bas"

Const As Double Euler = 0.57721566490153286

Dim As Double m = 0
For x As Ulongint = 2 To 1e8
If isPrime(x) Then m += Log(1-(1/x)) + (1/x)
Next x
Print "MM ="; Euler + m
Sleep
```
Output:
`MM = 0.2614972131104154`

## Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"math"
"rcu"
)

func contains(a []int, f int) bool {
for _, e := range a {
if e == f {
return true
}
}
return false
}

func main() {
const euler = 0.57721566490153286
primes := rcu.Primes(1 << 31)
pc := len(primes)
sum := 0.0
fmt.Println("------------  --------------")
for i, p := range primes {
rp := 1.0 / float64(p)
sum += math.Log(1.0-rp) + rp
c := i + 1
if (c%1e7) == 0 || c == pc {
fmt.Printf("%11s   %0.12f\n", rcu.Commatize(c), sum+euler)
}
}
}
```
Output:
```Primes added         M
------------  --------------
10,000,000   0.261497212987
20,000,000   0.261497212912
30,000,000   0.261497212889
40,000,000   0.261497212878
50,000,000   0.261497212871
60,000,000   0.261497212867
70,000,000   0.261497212864
80,000,000   0.261497212862
90,000,000   0.261497212861
100,000,000   0.261497212859
105,097,565   0.261497212858
```

## J

Implementation

```Euler=: 0.57721566490153286
MM=: {{ Euler + +/ (+ ^.@-.)@% p: i. y }}
```

```   0j13 ": MM 1e8
0.2614972128591
```

That said, a more literal implementation, like

```mm=: (% 10x(^#)10#.inv]) 26149721284764278375542683860869585905156664826119920619206421392x
```

would be more precise:

```   0j65":mm
0.26149721284764278375542683860869585905156664826119920619206421392
```

## Java

```import java.util.ArrayList;
import java.util.BitSet;
import java.util.List;

public final class MeisselMertensConstant {

public static void main(String[] aArgs) {
List<Double> primeReciprocals = listPrimeReciprocals(1_000_000_000);
final double euler = 0.577_215_664_901_532_861;
double sum = 0.0;
for ( double reciprocal : primeReciprocals ) {
sum += reciprocal + Math.log(1.0 - reciprocal);
}

final double meisselMertens = euler + sum;
System.out.println(String.format("%s%.9f", "The Meissel-Mertens constant = ", meisselMertens));
}

private static List<Double> listPrimeReciprocals(int aLimit) {
BitSet sieve = new BitSet(aLimit + 1);
sieve.set(2, aLimit + 1);

for ( int i = 2; i <= Math.sqrt(aLimit); i = sieve.nextSetBit(i + 1) ) {
for ( int j = i * i; j <= aLimit; j += i ) {
sieve.clear(j);
}
}

List<Double> result = new ArrayList<Double>(sieve.cardinality());
for ( int i = 2; i >= 0; i = sieve.nextSetBit(i + 1) ) {
}

return result;
}

}
```
Output:
```The Meissel-Mertens constant = 0.261497213
```

## Julia

Off by one in the 11th digit after 10^8 primes.

```using Base.MathConstants  # sets constant γ = 0.5772156649015...
using Primes

""" Approximate the Meissel-Mertons constant. """
function meissel_mertens(iterations = 100_000_000)
return mapreduce(p ->(d = 1/p; log(1 - d) + d), +, primes(prime(iterations))) + γ
end

@show meissel_mertens(100_000_000) # meissel_mertens(100000000) = 0.2614972128591237
```

## Maxima

```meissel_mertens:%gamma+lsum(log(1-(1/i))+(1/i),i,primes(2,10000000)),numer;
```
Output:
```0.2614972157776471
```

## Nim

Translation of: Wren
```import std/[math, strformat, strutils]

proc initPrimes(N: static int): seq[int] =
## Initialize the list of primes.

const M = 2 * N - 1
var composite = newSeq[bool](N)
composite[0] = true   # 1 is not prime.

# Conversions from index to value and value to index.
template index(n: int): int = (n - 1) shr 1
template value(idx: int): int = idx shl 1 + 1

# Fill the sieve.
var n = 3
while n * n <= M:
if not composite[n.index]:
for k in countup(n * n, M, 2 * n):
composite[k.index] = true
inc n, 2

# Build list of primes.
result = @[2]
for idx in 0..composite.high:
if not composite[idx]:

const N = 2^30
let primes = initPrimes(N)

echo "────────────   ──────────────"
const γ = 0.57721566490153286   # Euler–Mascheroni constant.
let primeCount = primes.len
var sum = 0.0
var count = 0
for p in primes:
let rp = 1 / p
sum += ln(1 - rp) + rp
inc count
if count mod 10_000_000 == 0 or count == primeCount:
echo &"{insertSep(\$count):>11}   {sum+γ:.12}"
```
Output:
```Primes added         M
────────────   ──────────────
10_000_000   0.261497212987
20_000_000   0.261497212912
30_000_000   0.261497212889
40_000_000   0.261497212878
50_000_000   0.261497212871
60_000_000   0.261497212867
70_000_000   0.261497212864
80_000_000   0.261497212862
90_000_000   0.261497212861
100_000_000   0.261497212859
105_097_565   0.261497212858
```

## PARI/GP

### Summation method

```{
MM(t)=
my(s=0);
forprime(p = 2, t,
s += log(1.-1./p)+1./p
);
Euler+s
};```
Output:

10 valid digits, prime summation up to 10^9.

```? \p 12
realprecision = 19 significant digits (12 digits displayed)
? MM(1e9)
?
%1 = 0.261497212874
?
? ##
***   last result: cpu time 1min, 18,085 ms, real time 1min, 18,094 ms.
?```

### Analytic method

The Analytic method requires high precision calculation of Riemann zeta function.

```{
Meissel_Mertens(d)=
default(realprecision, d);
my(prec = default(realprecision), z = 0, y = 0, q);
forprime(p = 2 , 7,
z += log(1.-1./p)+1./p
);
for(k = 2, prec,
q = 1;
forprime(p = 2, 7,
q *= 1.-p^-k
);
y += moebius(k)*log(zeta(k)*q)/k
);
Euler+z+y
};```
Output:

1000 valid digits. For some reason the last digit is in some rare cases (rounded) wrong. Whatever, the last digit is per definition the one with the lowest weight.

```? Meissel_Mertens(1001)
%1 = 0.26149721284764278375542683860869585905156664826119920619206421392492451089736820971414263143424665105161772887648602199778339032427004442454348740197238640666194955709392581712774774211985258807266272064144464232590023543105177232173925663229980314763831623758149059290382284758265972363422015971458785446941586825460538918007031787714156680620570605257601785334398970354507934530971953511716888598019955346947142883673537117910619342522616975101911159537244599605203558051780574237201332999961769676911386909654186249097435916294862238555389898241954857937738258646582212506260380084370067541379219020626760709633535981989783010762417792511961619355361391684002933280522289185167238258837930443067100391254985761418536020400457460311825670423438456551983202200477824746954606715454777572171338072595463648319687279859427306787306509669454587505942593547068846408425666008833035029366514525328713339609172639368543886291288200447611698748441593459920236225093315001729474600911978170842383659092665509
? ##
***   last result: cpu time 283 ms, real time 284 ms.```

## Perl

Library: ntheory
```use v5.36;
use ntheory qw(forprimes);

my \$s;
forprimes { \$s += log(1 - 1/\$_)+1/\$_ } 1e9;
say my \$result = \$s + .57721566490153286;
```
Output:
`0.261497212871049`

## Phix

Converges very slowly, I started running out of memory (from generating so many primes) before it showed any signs of getting stuck.

```with javascript_semantics -- (but perhaps a bit too slow)
constant mmc = 0.2614972128476427837554268386086958590516,
smmc = "0.2614972128476427837554268386086958590516"
atom t = 0.57721566490153286, dpa = 1, p10=1
integer pn = 1, adp = 0
string st = "0", fmt = "%.0f"
printf(1,"------------  --------------\n")
atom rp = 1/get_prime(pn)
t += log(1-rp) + rp
if (t-mmc)<dpa then
string ft = sprintf(fmt,trunc(t*p10)/p10) -- (as below)
if ft=st then
--
-- We have to artifically calculate a "truncated t", aka tt,
-- to prevent say 0.2..299[>5..] being automatically rounded
-- by printf() to 0.2..300, otherwise it just "looks wrong".
--
atom tt = trunc(t*1e12)/1e12
printf(1,"%,11d   %0.12f (accurate to %d d.p.)\n", {pn, tt, adp})
dpa /= 10
p10 *= 10
end if
end if
pn += 1
end while
printf(1,"(actual value 0.26149721284764278375542683860869)\n")
```
Output:

(Couldn't be bothered with making it an inner loop to upgrade the "accurate to 4dp" to 5dp)

```Primes added         M
------------  --------------
1   0.384068484341 (accurate to 0 d.p.)
3   0.288793158252 (accurate to 1 d.p.)
6   0.269978901636 (accurate to 2 d.p.)
38   0.261990332075 (accurate to 3 d.p.)
1,940   0.261499998883 (accurate to 4 d.p.)
1,941   0.261499997116 (accurate to 5 d.p.)
5,471   0.261497999951 (accurate to 6 d.p.)
34,891   0.261497299999 (accurate to 7 d.p.)
303,447   0.261497219999 (accurate to 8 d.p.)
9,246,426   0.261497212999 (accurate to 9 d.p.)
24,304,615   0.261497212899 (accurate to 10 d.p.)
(actual value 0.26149721284764278375542683860869)
```

### analytical/gmp

Translation of: PARI/GP
```without js -- no mpfr_zeta[_ui]() in mpfr.js, as yet, or mpfr_log() or mpfr_const_euler(), for that matter
include mpfr.e
function moebius(integer n)
if n=1 then return 1 end if
sequence f = prime_factors(n,true)
for i=2 to length(f) do
if f[i] = f[i-1] then return 0 end if
end for
return iff(odd(length(f))?-1:+1)
end function

function Meissel_Mertens(integer d)
mpfr_set_default_precision(-d) -- (d decimal places)
mpfr {res,rp,z,y,q} = mpfr_inits(5)
for p in {2,3,5,7} do
-- z += log(1-1/p)+1/p
mpfr_set_si(rp,1)
mpfr_div_si(rp,rp,p)
mpfr_si_sub(rp,1,rp)
mpfr_log(rp,rp)
end for
for k=2 to d do -- (see note)
integer m = moebius(k)
if m then
mpfr_set_si(q,1)
for p in {2,3,5,7} do
-- q *= 1-power(p,-k)
mpfr_set_si(rp,p)
mpfr_pow_si(rp,rp,-k)
mpfr_si_sub(rp,1,rp)
mpfr_mul(q,q,rp)
end for
-- y += moebius(k)*log(zeta(k)*q)/k
mpfr_zeta_ui(rp,k)
mpfr_mul(rp,rp,q)
mpfr_log(rp,rp)
mpfr_div_si(rp,rp,k)
mpfr_mul_si(rp,rp,m)
end if
end for
-- res := EULER+z+y
mpfr_const_euler(res)
return res
end function

mpfr m = Meissel_Mertens(1001)
?shorten(mpfr_get_str(m,10,1001))
```
Output:

Agrees with PARI/GP except for the very last digit being 8 instead of 9. Note that playing with precision and/or iterations (which seems to differ quite wildly) gave utterly incorrect results... not that I actually comprehend what the algorithm is doing. The 1,003 includes 2 from the "0.", fairly obviously.

```"0.261497212847642783...70842383659092665508 (1,003 digits)"
```

## Python

Translation of: FreeBASIC
```#!/usr/bin/python

from math import log

def isPrime(n):
for i in range(2, int(n**0.5) + 1):
if n % i == 0:
return False
return True

if __name__ == '__main__':
Euler = 0.57721566490153286
m = 0
for x in range(2, 10_000_000):
if isPrime(x):
m += log(1-(1/x)) + (1/x)

print("MM =", Euler + m)
```
Output:
`MM = 0.26149721577764207`

## Raku

Translation of: FreeBASIC
```# 20221011 Raku programming solution

my \$s;
.is-prime and \$s += log(1-1/\$_)+1/\$_ for 2 .. 10**8;
say \$s + .57721566490153286
```
Output:
`0.26149721310571383`

## REXX

Libraries: How to use
Library: Numbers
Library: Functions
Library: Constants
Library: Sequences
Below program shows 3 methods to calculate the Meissel-Mertens constant.

• BruteForce: Check all odd integers 3...999999 on primality (Miller-Rabin) and use defining sum.
• UsingSieve: Get all primes below 1000000 (Sieve) and use defining sum.
• Analytic: Fast converging sum formula given by Wolfram MathWorld.
```arg n; if n = '' then n = 16; numeric digits n
parse version version; say version n 'digits'; glob. = ''; fact. = 0
say 'Meissel-Mertens constant'
say
call Time('r'); a = BruteForce(); e = Format(Time('e'),,3)
say 'BruteForce' a '('e 'seconds)'
call Time('r'); a = UsingSieve(); e = Format(Time('e'),,3)
say 'UsingSieve' a '('e 'seconds)'
call Time('r'); a = Analytic(); e = Format(Time('e'),,3)
say 'Analytic  ' a '('e 'seconds)'
call Time('r'); a = TrueValue(); e = Format(Time('e'),,3)
say 'True value' a '('e 'seconds)'
exit

BruteForce:
procedure expose glob.
numeric digits Digits()+2
y = 0.5-Ln(2)
do n = 3 by 2 to 1000000
if Prime(n) then do
q = 1/n; t = Ln(1-q)+q; y = y+t
end
end
y = Euler()+y
numeric digits Digits()-2
return y+0

UsingSieve:
procedure expose glob. prim.
numeric digits Digits()+2
n = Primes(1000000); y = 0
do i = 1 to n
q = 1/prim.prime.i; t = Ln(1-q)+q; y = y+t
end
y = Euler()+y
numeric digits Digits()-2
return y+0

Analytic:
procedure expose glob. fact.
numeric digits Digits()+2
y = 0; v = 0
do n = 2 to 1000
t = Moebius(n) * Ln(Zeta(n)) / n
if t <> 0 then do
y = y+t
if y = v then
leave
v = y
end
end
y = Euler()+y
numeric digits Digits()-2
return y+0

TrueValue:
procedure
return 0.261497212847642783755426838608695859051566648261199206192064213924924510897368209714142631434246651051617+0

include Constants
include Functions
include Numbers
```
Output:
```
Runs with 16 digits:
REXX-ooRexx_5.0.0(MT)_64-bit 6.05 23 Dec 2022 16 digits
Meissel-Mertens constant

BruteForce 0.2614972467355791 (19.263 seconds)
UsingSieve 0.2614972467355791 (1.510 seconds)
Analytic   0.2614972128476428 (0.040 seconds)
True value 0.2614972128476428 (0.000 seconds)

Runs with 111 digits:
REXX-ooRexx_5.0.0(MT)_64-bit 6.05 23 Dec 2022 111  digits
Meissel-Mertens constant

BruteForce 0.261497246735500132677260976437201892720087109609945002636762560229390985475526828746177041776354307483184476679 (57.470 seconds)
UsingSieve 0.261497246735500132677260976437201892720087109609945002636762560229390985475526828746177041776354307483184476679 (1.830 seconds)
Analytic   0.261497212847642783755426838608695859051566648261199206192064213924924510897368209714142631434246651051617728876 (48.760 seconds)
True value 0.261497212847642783755426838608695859051566648261199206192064213924924510897368209714142631434246651051617 (0.000 seconds)
```

Using primes only 7 correct decimals are achieved. UsingSieve is faster, but requires extra memory to store all primes.
In spite of the expensive functions Moebius and Zeta, the analytic solution outperforms. The Analytic results suggest that the OEIS entry is truncated, not rounded.

## RPL

`BPRIM?` is defined at Primality by trial division#RPL

Works with: Halcyon Calc version 4.2.8
```≪ → n
≪ 0.5 3 n FOR j
IF j R→B BPRIM? THEN j INV + END 2 STEP
n LN LN -
≫ ≫ 'MMCON' STO
```
```10000 MMCON
```
Output:
```1: 0.262733140866
```

## Sidef

```var sum = 0
1e7.each_prime {|p|
with (1f/p) {|t|
sum += (log(1 - t) + t)
}
}
say sum+Num.EulerGamma
```
Output:
```0.26149721577767111119422410228297206467931376306
```

## Wren

### Summation method

Library: Wren-math
Library: Wren-fmt

It will be seen that this is converging to the correct answer though we'd need to add a lot more primes to obtain a valid 11th digit.

```import "./math" for Int
import "./fmt" for Fmt

var euler = 0.57721566490153286
var primes = Int.primeSieve(2.pow(31))
var pc = primes.count
var sum = 0
var c = 0
System.print("------------  --------------")
for (p in primes) {
var rp = 1/p
sum = (1-rp).log + rp + sum
c = c + 1
if ((c % 1e7) == 0 || c == pc) Fmt.print("\$,11d   \$0.12f", c, sum + euler)
}
```
Output:
```Primes added         M
------------  --------------
10,000,000   0.261497212987
20,000,000   0.261497212912
30,000,000   0.261497212889
40,000,000   0.261497212878
50,000,000   0.261497212871
60,000,000   0.261497212867
70,000,000   0.261497212864
80,000,000   0.261497212862
90,000,000   0.261497212861
100,000,000   0.261497212859
105,097,565   0.261497212858
```

### Analytic method

Translation of: PARI/GP
Library: Wren-gmp

This agrees with the Phix entry that the 1,000th digit after the decimal point is '8' after rounding (according to OEIS it's actually '7' but the next one is '7' also).

```import "./gmp" for Mpf
import "./math" for Int
import "./fmt" for Fmt

var isSquareFree = Fn.new { |n|
var i = 2
while (i * i <= n) {
if (n%(i*i) == 0) return false
i = (i > 2) ? i + 2 : i + 1
}
return true
}

var mu = Fn.new { |n|
if (n < 1) Fiber.abort("Argument must be a positive integer")
if (n == 1) return 1
var sqFree = isSquareFree.call(n)
var factors = Int.primeFactors(n)
if (sqFree && factors.count % 2 == 0) return 1
if (sqFree) return -1
return 0
}

var meisselMertens = Fn.new { |d|
Mpf.defaultPrec = d
var z = Mpf.zero
var y = Mpf.zero
var r = Mpf.new()
var q = Mpf.new()
var t = Mpf.new()
var m = Mpf.new()
for (p in [2, 3, 5, 7]) {
r.setUi(p).inv
}
for (k in 2..d) {
q.setUi(1)
for (p in [2, 3, 5, 7]) {
r.setUi(p).inv
t.uiSub(1, r.pow(k))
q.mul(t)
}
m.setSi(mu.call(k))
t.zetaUi(k).mul(q).log.mul(m).div(k)
}
}

Fmt.print("\$20a", meisselMertens.call(3300).toString(1001))
```
Output:
```0.261497212847642783...70842383659092665508
```

## XPL0

Runs in 98 seconds on Pi4.

```func IsPrime(N); \Return 'true' if N is prime
int  N, D;
[if N < 2 then return false;
if (N&1) = 0 then return N = 2;
if rem(N/3) = 0 then return N = 3;
D:= 5;
while D*D <= N do
[if rem(N/D) = 0 then return false;
D:= D+2;
if rem(N/D) = 0 then return false;
D:= D+4;
];
return true;
];

def  Euler = 0.57721566490153286;
real M;  int  P;
[M:= 0.;
for P:= 2 to 100_000_000 do
if IsPrime(P) then
M:= M + Ln(1. - 1./float(P)) + 1./float(P);
Format(1, 16);
Text(0, "MM = ");  RlOut(0, Euler + M);  CrLf(0);
]```
Output:
```MM = 0.2614972131104150
```

## Mathematica / Wolfram Language

```PrimeNumbers = Select[Range[100000000], PrimeQ[#] &]; (*all primes in the first 100 000 000 numbers, this takes a toll on my computer's CPU and RAM*)
MM = N[Total[Log[1 - 1/PrimeNumbers] + 1/PrimeNumbers] + EulerGamma, 10] (*Calculating it up to a precision of 10, this is correct up to 8 digits*)
AnalyticMMto305 = N[EulerGamma + Sum[MoebiusMu[n]/n Log[Zeta[n]], {n, 2, 1000}], 1000] (*Precise up to 305 digits*)
AnalyticMM = N[EulerGamma + Sum[MoebiusMu[n]/n Log[Zeta[n]], {n, 2, 10000}], 1001] (*Precise up to at least 1000 digits*)
```
Output:
```0.2614972131
0.2614972128476427837554268386086958590515666482611992061920642139249245108973682097141426314342466510516177288764860219977833903242700444245434874019723864066619495570939258171277477421198525880726627206414446423259002354310517723217392566322998031476383162375814905929038228475826597236342201597145878545286546864785868143588282690675811212349253390275898516648963722862532834592270193787663814788872688899729598789511243611212105643183719483028350694889140412496558830385604700978671612478906659270558671778566494523172201279234383779159082832574101983245266507469788929642499636163109342230499137582782542115136325204939772977424363327903550519767079810438569344262666067369673777630199473914034464437524475856770642386781282897163202273071362276688321001599406964213181471899136557333922267566346325514106895289284293459552062339943989806700523705145085201774716630458161117596603698964857884739475619061322196278674275162487770072794776455644351959504021182511495417580620912301758280597008866059
0.26149721284764278375542683860869585905156664826119920619206421392492451089736820971414263143424665105161772887648602199778339032427004442454348740197238640666194955709392581712774774211985258807266272064144464232590023543105177232173925663229980314763831623758149059290382284758265972363422015971458785446941586825460538918007031787714156680620570605257601785334398970354507934530971953511716888598019955346947142883673537117910619342522616975101911159537244599605203558051780574237201332999961769676911386909654186249097435916294862238555389898241954857937738258646582212506260380084370067541379219020626760709633535981989783010762417792511961619355361391684002933280522289185167238258837930443067100391254985761418536020400457460311825670423438456551983202200477824746954606715454777572171338072595463648319687279859427306787306509669454587505942593547068846408425666008833035029366514525328713339609172639368543886291288200447611698748441593459920236225093315001729474600911978170842383659092665508

```