page_title: Meissel–Mertens constant
hierarchical_section_title: Meissel–Mertens constant
caption_attribution_description: English: The plot of the prime harmonic sum up to n = 2 15 , 2 16 , &#x2026; , 2 46 = 7.04 &#x00D7; 10 13 {\displaystyle n=2^{15},2^{16},\ldots ,2^{46}=7.04\times 10^{13}} and the Merten's approximation to it. The original of this figure has y {\displaystyle y} axis of the length 8 cm and spans the interval (2.5, 3.8), so if the n {\displaystyle n} axis would be plotted in the linear scale instead of logarithmic, then it should be 5.33 ( 3 ) &#x00D7; 10 9 {\displaystyle 5.33(3)\times 10^{9}} km long --- that is the size of the Solar System.
context_page_description: The Meissel–Mertens constant, also referred to as Mertens constant, Kronecker's constant, Hadamard–de la Vallée-Poussin constant or the prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm:
  Here γ is the Euler–Mascheroni constant, which has an analogous definition involving a sum over all integers.
  The value of M is approximately
  M ≈ 0.2614972128476427837554268386086958590516....
  Mertens' second theorem establishes that the limit exists.
  The fact that there are two logarithms in the limit for the Meissel–Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler–Mascheroni constant.
context_section_description: The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard–de la Vallée-Poussin constant or the prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm:
  Here γ is the Euler–Mascheroni constant, which has an analogous definition involving a sum over all integers (not just the primes).
  The value of M is approximately
  M ≈ 0.2614972128476427837554268386086958590516... (sequence A077761 in the OEIS).
  Mertens' second theorem establishes that the limit exists.
  The fact that there are two logarithms (log of a log) in the limit for the Meissel–Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler–Mascheroni constant.
caption: The plot of the prime harmonic sum up to 
    
      
        
          n
          =
          
            2
            
              15
            
          
          ,
          
            2
            
              16
            
          
          ,
          …
          ,
          
            2
            
              46
            
          
          ≈
          7.04
          ×
          
            10
            
              13
            
          
        
      
      {\displaystyle n=2^{15},2^{16},\ldots ,2^{46}\approx 7.04\times 10^{13}}
    
   and the Merten's approximation to it. The original of this figure has y axis of the length 8 cm and spans the interval (2.5, 3.8), so if the n axis would be plotted in the linear scale instead of logarithmic, then it should be 
    
      
        
          5.33
          (
          3
          )
          ×
          
            10
            
              9
            
          
        
      
      {\displaystyle 5.33(3)\times 10^{9}}
    
   km long — that is the size of the Solar System.
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