Buchstab function

<h2 id="asymptotics">Asymptotics</h2>
<p>The Buchstab function approaches 
  
    
      
        
          e
          
            −
            γ
          
        
        ≈
        0.561
      
    
    {\displaystyle e^{-\gamma }\approx 0.561}
  
 rapidly as 
  
    
      
        u
        →
        ∞
        ,
      
    
    {\displaystyle u\to \infty ,}
  
 where 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 is the <a href="/facts/Euler%E2%80%93Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>.  In fact,
</p>

|
        
        ω
        (
        u
        )
        −
        
          e
          
            −
            γ
          
        
        
          |
        
        ≤
        
          
            
              ρ
              (
              u
              −
              1
              )
            
            u
          
        
        ,
        
        u
        ≥
        1
        ,
      
    
    {\displaystyle |\omega (u)-e^{-\gamma }|\leq {\frac {\rho (u-1)}{u}},\qquad u\geq 1,}

<p>where <i>ρ</i>  is the <a href="/facts/Dickman_function/PE99PvRS">Dickman function</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>  Also, 
  
    
      
        ω
        (
        u
        )
        −
        
          e
          
            −
            γ
          
        
      
    
    {\displaystyle \omega (u)-e^{-\gamma }}
  
 oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima.  The interval between consecutive extrema approaches 1 as <i>u</i> approaches infinity, as does the interval between consecutive zeroes.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="applications">Applications</h2>
<p>The Buchstab function is used to count <a href="/facts/Rough_number/63cADSmy">rough numbers</a>.
If Φ(<i>x</i>, <i>y</i>) is the number of positive integers less than or equal to <i>x</i> with no prime factor less than <i>y</i>, then for any fixed <i>u</i> > 1,
</p>

Φ
        (
        x
        ,
        
          x
          
            1
            
              /
            
            u
          
        
        )
        ∼
        ω
        (
        u
        )
        
          
            x
            
              log
              ⁡
              
                x
                
                  1
                  
                    /
                  
                  u
                
              
            
          
        
        ,
        
        x
        →
        ∞
        .
      
    
    {\displaystyle \Phi (x,x^{1/u})\sim \omega (u){\frac {x}{\log x^{1/u}}},\qquad x\to \infty .}

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Alexander_Buchstab/tE0H0PQR">Бухштаб, А. А.</a> (1937), <a href="https://mi.mathnet.ru/msb5649">"Асимптотическая оценка одной общей теоретикочисловой функции"</a> [Asymptotic estimation of a general number-theoretic function], <i><a href="/facts/Matematicheskii_Sbornik/fPeQKbDx">Matematicheskii Sbornik</a></i> (in Russian), 2(44) (6): 1239–1246, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0018.24504">0018.24504</a></li>
<li><a href="http://mathworld.wolfram.com/BuchstabFunction.html">"Buchstab Function"</a>, <a href="/facts/Wolfram_MathWorld/yc1jodbq">Wolfram MathWorld</a>.  Accessed on line Feb. 11, 2015.</li>
<li>§IV.32, "On Φ(x,y) and Buchstab's function", <i>Handbook of Number Theory I</i>, József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer</a>, 2006, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4020-4215-7.</li>
<li><a href="https://www.ams.org/journals/mcom/1990-55-191/S0025-5718-1990-1023043-8/">"A differential delay equation arising from the sieve of Eratosthenes"</a>, A. Y. Cheer and D. A. Goldston, <i><a href="/facts/Mathematics_of_Computation/KAFdf8sw">Mathematics of Computation</a></i> 55 (1990), pp. 129–141.</li>
<li>"An improvement of Selberg’s sieve method", W. B. Jurkat and H.-E. Richert, <i><a href="/facts/Acta_Arithmetica/9hoeHS3j">Acta Arithmetica</a></i> 11 (1965), pp. 217–240.</li>
<li>Hildebrand, A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Bukhstab_function">"Bukhstab function"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>(5.13), Jurkat and Richert 1965.  In this paper the argument of ρ has been shifted by 1 from the usual definition. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>p. 131, Cheer and Goldston 1990. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Buchstab function open-in-new

Buchstab function