Divergence of the sum of the reciprocals of the primes

The sum of the <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocals</a> of all <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> <a href="/facts/Divergent_series/c8Vjjjdx">diverges</a>; that is:

∑
          
            p
            
               prime
            
          
        
        
          
            1
            p
          
        
        =
        
          
            1
            2
          
        
        +
        
          
            1
            3
          
        
        +
        
          
            1
            5
          
        
        +
        
          
            1
            7
          
        
        +
        
          
            1
            11
          
        
        +
        
          
            1
            13
          
        
        +
        
          
            1
            17
          
        
        +
        ⋯
        =
        ∞
      
    
    {\displaystyle \sum _{p{\text{ prime}}}{\frac {1}{p}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+\cdots =\infty }

This was proved by <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> in 1737, and strengthens <a href="/facts/Euclid/dHpVyL4R">Euclid</a>'s 3rd-century-BC result that <a href="/facts/Euclid%2527s_theorem/Nl4SM0P0">there are infinitely many prime numbers</a> and <a href="/facts/Nicole_Oresme/sv3C8R2s">Nicole Oresme</a>'s 14th-century proof of the <a href="/facts/Divergence_of_the_harmonic_series/8KGDHKTg">divergence of the sum of the reciprocals of the integers (harmonic series)</a>.
There are a variety of proofs of Euler's result, including a <a href="/facts/Lower_bound/YJje9oC2">lower bound</a> for the partial sums stating that

∑
          
            
              
                p
                
                   prime
                
              
              
                p
                ≤
                n
              
            
          
        
        
          
            1
            p
          
        
        ≥
        log
        ⁡
        log
        ⁡
        (
        n
        +
        1
        )
        −
        log
        ⁡
        
          
            
              π
              
                2
              
            
            6
          
        
      
    
    {\displaystyle \sum _{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq n}{\frac {1}{p}}\geq \log \log(n+1)-\log {\frac {\pi ^{2}}{6}}}

for all natural numbers n. The double <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a> (log log) indicates that the divergence might be very slow, which is indeed the case. See <a href="/facts/Meissel%25E2%2580%2593Mertens_constant/AEsCfwH5">Meissel–Mertens constant</a>.

Divergence of the sum of the reciprocals of the primes open-in-new

Divergence of the sum of the reciprocals of the primes