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Meissel–Mertens constant

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


Task

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
Memo.Lines.Add('Primes  Digits       M');
Memo.Lines.Add('-----------------------------------------');
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))));
	Memo.Lines.Add(Format('10^%2d  %7d  %25.18f',[I,Digits,M]));
	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("Primes added         M")
    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 }}

Task example

   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) ) {
			result.add(1.0 / i);
		}
		
		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]:
      result.add idx.value


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

echo "Primes added         M"
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,"Primes added         M\n")
printf(1,"------------  --------------\n")
while adp<=10 do
    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})
            adp += 1
            fmt = sprintf("%%.%df",adp)
            st = smmc[1..2+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_add(z,z,rp)
        mpfr_si_sub(rp,1,rp)
        mpfr_log(rp,rp)
        mpfr_add(z,z,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)
            mpfr_add(y,y,rp)
        end if
    end for
    -- res := EULER+z+y
    mpfr_const_euler(res)
    mpfr_add(z,z,y)
    mpfr_add(res,res,z)
    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: Settings
Library: Abend
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.
include Settings

say version; say 'Meissel-Mertens constant'; say
arg n
if n = '' then
   n = 16
numeric digits n
fact. = 0
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
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("Primes added         M")
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
        t.uiSub(1, r).log.add(r)
        z.add(t, z)
    }
    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)
        y.add(t, y)
    }
    return Mpf.euler.add(z).add(y)
}

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

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