Mertens function

<h2 id="representations">Representations</h2>
<h3>As an integral</h3>
Using the <a href="/facts/Euler_product/Lvsv4U3M">Euler product</a>, one finds that

1
            
              ζ
              (
              s
              )
            
          
        
        =
        
          ∏
          
            p
          
        
        (
        1
        −
        
          p
          
            −
            s
          
        
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              μ
              (
              n
              )
            
            
              n
              
                s
              
            
          
        
        ,
      
    
    {\displaystyle {\frac {1}{\zeta (s)}}=\prod _{p}(1-p^{-s})=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}},}

where 
 
 
 
 ζ
 (
 s
 )
 
 
 {\displaystyle \zeta (s)}
 
 is the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>, and the product is taken over primes. Then, using this <a href="/facts/Dirichlet_series/uIAUaLcd">Dirichlet series</a> with <a href="/facts/Perron%2527s_formula/Decmf4Y2">Perron's formula</a>, one obtains

1
            
              2
              π
              i
            
          
        
        
          ∫
          
            c
            −
            i
            ∞
          
          
            c
            +
            i
            ∞
          
        
        
          
            
              x
              
                s
              
            
            
              s
              ζ
              (
              s
              )
            
          
        
        
        d
        s
        =
        M
        (
        x
        )
        ,
      
    
    {\displaystyle {\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {x^{s}}{s\zeta (s)}}\,ds=M(x),}

where c > 1.
Conversely, one has the <a href="/facts/Mellin_transform/M0087pMe">Mellin transform</a>

1
            
              ζ
              (
              s
              )
            
          
        
        =
        s
        
          ∫
          
            1
          
          
            ∞
          
        
        
          
            
              M
              (
              x
              )
            
            
              x
              
                s
                +
                1
              
            
          
        
        
        d
        x
        ,
      
    
    {\displaystyle {\frac {1}{\zeta (s)}}=s\int _{1}^{\infty }{\frac {M(x)}{x^{s+1}}}\,dx,}

which holds for 
 
 
 
 Re
 ⁡
 (
 s
 )
 >
 1
 
 
 {\displaystyle \operatorname {Re} (s)>1}
 
.
A curious relation given by Mertens himself involving the second <a href="/facts/Chebyshev_function/Kufg2P2S">Chebyshev function</a> is

ψ
        (
        x
        )
        =
        M
        
          (
          
            
              x
              2
            
          
          )
        
        log
        ⁡
        2
        +
        M
        
          (
          
            
              x
              3
            
          
          )
        
        log
        ⁡
        3
        +
        M
        
          (
          
            
              x
              4
            
          
          )
        
        log
        ⁡
        4
        +
        ⋯
        .
      
    
    {\displaystyle \psi (x)=M\left({\frac {x}{2}}\right)\log 2+M\left({\frac {x}{3}}\right)\log 3+M\left({\frac {x}{4}}\right)\log 4+\cdots .}

Assuming that the Riemann zeta function has no multiple non-trivial zeros, one has the "exact formula" by the <a href="/facts/Residue_theorem/M5ITFIyx">residue theorem</a>:

M
        (
        x
        )
        =
        
          ∑
          
            ρ
          
        
        
          
            
              x
              
                ρ
              
            
            
              ρ
              
                ζ
                ′
              
              (
              ρ
              )
            
          
        
        −
        2
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              (
              −
              1
              
                )
                
                  n
                  −
                  1
                
              
              (
              2
              π
              
                )
                
                  2
                  n
                
              
            
            
              (
              2
              n
              )
              !
              n
              ζ
              (
              2
              n
              +
              1
              )
              
                x
                
                  2
                  n
                
              
            
          
        
        .
      
    
    {\displaystyle M(x)=\sum _{\rho }{\frac {x^{\rho }}{\rho \zeta '(\rho )}}-2+\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}(2\pi )^{2n}}{(2n)!n\zeta (2n+1)x^{2n}}}.}

<a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl</a> conjectured that the Mertens function satisfied the approximate functional-differential equation

y
              (
              x
              )
            
            2
          
        
        −
        
          ∑
          
            r
            =
            1
          
          
            N
          
        
        
          
            
              B
              
                2
                r
              
            
            
              (
              2
              r
              )
              !
            
          
        
        
          D
          
            t
          
          
            2
            r
            −
            1
          
        
        y
        
          (
          
            
              x
              
                t
                +
                1
              
            
          
          )
        
        +
        x
        
          ∫
          
            0
          
          
            x
          
        
        
          
            
              y
              (
              u
              )
            
            
              u
              
                2
              
            
          
        
        
        d
        u
        =
        
          x
          
            −
            1
          
        
        H
        (
        log
        ⁡
        x
        )
        ,
      
    
    {\displaystyle {\frac {y(x)}{2}}-\sum _{r=1}^{N}{\frac {B_{2r}}{(2r)!}}D_{t}^{2r-1}y\left({\frac {x}{t+1}}\right)+x\int _{0}^{x}{\frac {y(u)}{u^{2}}}\,du=x^{-1}H(\log x),}

where H(x) is the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>, B are <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli numbers</a>, and all derivatives with respect to t are evaluated at t = 0.
There is also a trace formula involving a sum over the Möbius function and zeros of the Riemann zeta function in the form

∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              μ
              (
              n
              )
            
            
              n
            
          
        
        g
        (
        log
        ⁡
        n
        )
        =
        
          ∑
          
            γ
          
        
        
          
            
              h
              (
              γ
              )
            
            
              
                ζ
                ′
              
              (
              1
              
                /
              
              2
              +
              i
              γ
              )
            
          
        
        +
        2
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              (
              −
              1
              
                )
                
                  n
                
              
              (
              2
              π
              
                )
                
                  2
                  n
                
              
            
            
              (
              2
              n
              )
              !
              ζ
              (
              2
              n
              +
              1
              )
            
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        g
        (
        x
        )
        
          e
          
            −
            x
            (
            2
            n
            +
            1
            
              /
            
            2
            )
          
        
        
        d
        x
        ,
      
    
    {\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{\sqrt {n}}}g(\log n)=\sum _{\gamma }{\frac {h(\gamma )}{\zeta '(1/2+i\gamma )}}+2\sum _{n=1}^{\infty }{\frac {(-1)^{n}(2\pi )^{2n}}{(2n)!\zeta (2n+1)}}\int _{-\infty }^{\infty }g(x)e^{-x(2n+1/2)}\,dx,}

where the first sum on the right-hand side is taken over the non-trivial zeros of the Riemann zeta function, and (g, h) are related by the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a>, such that

2
        π
        g
        (
        x
        )
        =
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        h
        (
        u
        )
        
          e
          
            i
            u
            x
          
        
        
        d
        u
        .
      
    
    {\displaystyle 2\pi g(x)=\int _{-\infty }^{\infty }h(u)e^{iux}\,du.}

<h3>As a sum over Farey sequences</h3>
Another formula for the Mertens function is

M
        (
        n
        )
        =
        −
        1
        +
        
          ∑
          
            a
            ∈
            
              
                
                  F
                
              
              
                n
              
            
          
        
        
          e
          
            2
            π
            i
            a
          
        
        ,
      
    
    {\displaystyle M(n)=-1+\sum _{a\in {\mathcal {F}}_{n}}e^{2\pi ia},}

where 
 
 
 
 
 
 
 F
 
 
 
 n
 
 
 
 
 {\displaystyle {\mathcal {F}}_{n}}
 
 is the <a href="/facts/Farey_sequence/P6n10vaU">Farey sequence</a> of order n.
This formula is used in the proof of the <a href="/facts/Farey_sequence/P6n10vaU">Franel–Landau theorem</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h3>As a determinant</h3>
M(n) is the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the n × n <a href="/facts/Redheffer_matrix/vyzzxqsR">Redheffer matrix</a>, a <a href="/facts/Logical_matrix/pS4BnBzj">(0, 1) matrix</a> in which
aij is 1 if either j is 1 or i divides j.

<h3>As a sum of the number of points under n-dimensional hyperboloids</h3>

M
        (
        x
        )
        =
        1
        −
        
          ∑
          
            2
            ≤
            a
            ≤
            x
          
        
        1
        +
        
          
            
              
                ∑
                
                  a
                  ≥
                  2
                
              
              
                ∑
                
                  b
                  ≥
                  2
                
              
            
            
              a
              b
              ≤
              x
            
          
        
        1
        −
        
          
            
              
                ∑
                
                  a
                  ≥
                  2
                
              
              
                ∑
                
                  b
                  ≥
                  2
                
              
              
                ∑
                
                  c
                  ≥
                  2
                
              
            
            
              a
              b
              c
              ≤
              x
            
          
        
        1
        +
        
          
            
              
                ∑
                
                  a
                  ≥
                  2
                
              
              
                ∑
                
                  b
                  ≥
                  2
                
              
              
                ∑
                
                  c
                  ≥
                  2
                
              
              
                ∑
                
                  d
                  ≥
                  2
                
              
            
            
              a
              b
              c
              d
              ≤
              x
            
          
        
        1
        −
        ⋯
      
    
    {\displaystyle M(x)=1-\sum _{2\leq a\leq x}1+{\underset {ab\leq x}{\sum _{a\geq 2}\sum _{b\geq 2}}}1-{\underset {abc\leq x}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}}}1+{\underset {abcd\leq x}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}\sum _{d\geq 2}}}1-\cdots }

This formulation expanding the Mertens function suggests asymptotic bounds obtained by considering the <a href="/facts/Divisor_summatory_function/LC52uv0F">Piltz divisor problem</a>, which generalizes the <a href="/facts/Divisor_summatory_function/LC52uv0F">Dirichlet divisor problem</a> of computing <a href="/facts/Asymptotic_estimate/kt87qLpu">asymptotic estimates</a> for the summatory function of the <a href="/facts/Divisor_function/rzvm1jw2">divisor function</a>.

<h2 id="other-properties">Other properties</h2>
From <a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> we have

∑
          
            d
            =
            1
          
          
            n
          
        
        M
        (
        ⌊
        n
        
          /
        
        d
        ⌋
        )
        =
        1
         
        .
      
    
    {\displaystyle \sum _{d=1}^{n}M(\lfloor n/d\rfloor )=1\ .}

Furthermore, from <a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

∑
          
            d
            =
            1
          
          
            n
          
        
        M
        (
        ⌊
        n
        
          /
        
        d
        ⌋
        )
        d
        =
        Φ
        (
        n
        )
         
        ,
      
    
    {\displaystyle \sum _{d=1}^{n}M(\lfloor n/d\rfloor )d=\Phi (n)\ ,}

where 
 
 
 
 Φ
 (
 n
 )
 
 
 {\displaystyle \Phi (n)}
 
 is the <a href="/facts/Totient_summatory_function/6xuABC8g">totient summatory function</a>.

<h2 id="calculation">Calculation</h2>
Neither of the methods mentioned previously leads to practical algorithms to calculate the Mertens function.
Using sieve methods similar to those used in prime counting, the Mertens function has been computed for all integers up to an increasing range of x.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<table><tbody><tr><th>Person</th><th>Year</th><th>Limit</th></tr><tr><td>Mertens</td><td>1897</td><td>104</td></tr><tr><td>von Sterneck</td><td>1897</td><td>1.5×105</td></tr><tr><td>von Sterneck</td><td>1901</td><td>5×105</td></tr><tr><td>von Sterneck</td><td>1912</td><td>5×106</td></tr><tr><td>Neubauer</td><td>1963</td><td>108</td></tr><tr><td>Cohen and Dress</td><td>1979</td><td>7.8×109</td></tr><tr><td>Dress</td><td>1993</td><td>1012</td></tr><tr><td>Lioen and van de Lune</td><td>1994</td><td>1013</td></tr><tr><td>Kotnik and van de Lune</td><td>2003</td><td>1014</td></tr><tr><td>Boncompagni</td><td>2011<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></td><td>1017</td></tr><tr><td>Kuznetsov</td><td>2012<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></td><td>1022</td></tr><tr><td>Helfgott and Thompson</td><td>2021<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></td><td>1023</td></tr></tbody></table>
The Mertens function for all integer values up to x may be computed in O(x log log x) time. A combinatorial algorithm has been developed incrementally starting in 1870 by <a href="/facts/Ernst_Meissel/xoWjVwv5">Ernst Meissel</a>,<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> <a href="/facts/Derrick_Henry_Lehmer/EI61SICa">Lehmer</a>,<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> <a href="/facts/Jeffrey_Lagarias/HVvI3c3E">Lagarias</a>-<a href="/facts/Victor_S._Miller/W25VdAe3">Miller</a>-<a href="/facts/Andrew_Odlyzko/rBsYtPz5">Odlyzko</a>,<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> and Deléglise-Rivat<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> that computes isolated values of M(x) in O(x2/3(log log x)1/3) time; a further improvement by <a href="/facts/Harald_Helfgott/CurPobx2">Harald Helfgott</a> and Lola Thompson in 2021 improves this to O(x3/5(log x)3/5+ε),<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> and an algorithm by Lagarias and Odlyzko based on integrals of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> achieves a running time of O(x1/2+ε).<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>
See <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A084237 for values of M(x) at powers of 10.

<h2 id="known-upper-bounds">Known upper bounds</h2>
Ng notes that the <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a> (RH) is equivalent to

M
        (
        x
        )
        =
        O
        
          (
          
            
              
                x
              
            
            exp
            ⁡
            
              (
              
                
                  
                    C
                    ⋅
                    log
                    ⁡
                    x
                  
                  
                    log
                    ⁡
                    log
                    ⁡
                    x
                  
                
              
              )
            
          
          )
        
        ,
      
    
    {\displaystyle M(x)=O\left({\sqrt {x}}\exp \left({\frac {C\cdot \log x}{\log \log x}}\right)\right),}

for some positive constant 
 
 
 
 C
 >
 0
 
 
 {\displaystyle C>0}
 
. Other upper bounds have been obtained by Maier, Montgomery, and Soundarajan assuming the RH including

|
                
                M
                (
                x
                )
                
                  |
                
              
              
                
                ≪
                
                  
                    x
                  
                
                exp
                ⁡
                
                  (
                  
                    
                      C
                      
                        2
                      
                    
                    ⋅
                    (
                    log
                    ⁡
                    x
                    
                      )
                      
                        
                          39
                          61
                        
                      
                    
                  
                  )
                
              
            
            
              
                
                  |
                
                M
                (
                x
                )
                
                  |
                
              
              
                
                ≪
                
                  
                    x
                  
                
                exp
                ⁡
                
                  (
                  
                    
                      
                        log
                        ⁡
                        x
                      
                    
                    (
                    log
                    ⁡
                    log
                    ⁡
                    x
                    
                      )
                      
                        14
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}|M(x)|&\ll {\sqrt {x}}\exp \left(C_{2}\cdot (\log x)^{\frac {39}{61}}\right)\\|M(x)|&\ll {\sqrt {x}}\exp \left({\sqrt {\log x}}(\log \log x)^{14}\right).\end{aligned}}}

Known explicit upper bounds without assuming the RH are given by:<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

|
 
 M
 (
 x
 )
 
 |
 
 
 
 
 <
 
 
 
 12590292
 ⋅
 x
 
 
 
 log
 
 236
 
 /
 
 75
 
 
 ⁡
 (
 x
 )
 
 
 
 ,
  
 
  for 
 
 x
 >
 exp
 ⁡
 (
 12282.3
 )
 
 
 
 
 
 |
 
 M
 (
 x
 )
 
 |
 
 
 
 
 <
 
 
 
 0.6437752
 ⋅
 x
 
 
 log
 ⁡
 x
 
 
 
 ,
  
 
  for 
 
 x
 >
 1.
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}|M(x)|&<{\frac {12590292\cdot x}{\log ^{236/75}(x)}},\ {\text{ for }}x>\exp(12282.3)\\|M(x)|&<{\frac {0.6437752\cdot x}{\log x}},\ {\text{ for }}x>1.\end{aligned}}}

It is possible to simplify the above expression into a less restrictive but illustrative form as:

M
                (
                x
                )
                =
                O
                
                  (
                  
                    
                      x
                      
                        
                          log
                          
                            π
                          
                        
                        ⁡
                        (
                        x
                        )
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}M(x)=O\left({\frac {x}{\log ^{\pi }(x)}}\right).\end{aligned}}}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Perron%2527s_formula/Decmf4Y2">Perron's formula</a></li>
<li><a href="/facts/Liouville%2527s_function/HCP4Ukr5">Liouville's function</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Harold_Edwards_(mathematician)/JKfNgfJB">Edwards, Harold</a> (1974). Riemann's Zeta Function. Mineola, New York: Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-41740-9.</li>
<li>Mertens, F. (1897). ""Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich". Kleine Sitzungsber, IIA. 106: 761–830.</li>
<li><a href="/facts/A._M._Odlyzko/rBsYtPz5">Odlyzko, A. M.</a>; <a href="/facts/Herman_te_Riele/AdNLnqwo">te Riele, Herman</a> (1985). <a href="http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf">"Disproof of the Mertens Conjecture"</a> (PDF). Journal für die reine und angewandte Mathematik. 357: 138–160.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/MertensFunction.html">"Mertens function"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="/facts/Neil_Sloane/kLWw29Me">Sloane, N. J. A.</a> (ed.). <a href="https://oeis.org/A002321">"Sequence A002321 (Mertens's function)"</a>. The <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</li>
<li>Deléglise, M. and Rivat, J. "Computing the Summation of the Möbius Function." Experiment. Math. 5, 291-295, 1996. <a href="https://projecteuclid.org/euclid.em/1047565447">Computing the summation of the Möbius function</a></li>
<li>Hurst, Greg (2016). "Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1610.08551">1610.08551</a> [<a href="https://arxiv.org/archive/math.NT">math.NT</a>].</li>
<li>Nathan Ng, "The distribution of the summatory function of the Möbius function", Proc. London Math. Soc. (3) 89 (2004) 361-389. <a href="http://www.cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf">[1]</a></li></ul>

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

<ol>
<li id="fn:1">Davenport, H. (November 1937). "On Some Infinite Series Involving Arithmetical Functions (Ii)". The Quarterly Journal of Mathematics. Original Series. 8 (1): 313–320. doi:10.1093/qmath/os-8.1.313. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Nathan Ng (October 25, 2018). "The distribution of the summatory function of the Mobius function". arXiv:math/0310381. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Edwards, Ch. 12.2. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lehman, R.S. (1960). "On Liouville's Function". Math. Comput. 14: 311–320. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Kanemitsu, S.; Yoshimoto, M. (1996). "Farey series and the Riemann hypothesis". Acta Arithmetica. 75 (4): 351–374. doi:10.4064/aa-75-4-351-374. <a href="https://doi.org/10.4064%2Faa-75-4-351-374" target="_blank">https://doi.org/10.4064%2Faa-75-4-351-374</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Kotnik, Tadej; van de Lune, Jan (November 2003). "Further systematic computations on the summatory function of the Möbius function". Modelling, Analysis and Simulation. MAS-R0313. <a href="https://ir.cwi.nl/pub/4116" target="_blank">https://ir.cwi.nl/pub/4116</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Hurst, Greg (2016). "Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture". arXiv:1610.08551 [math.NT]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Sloane, N. J. A. (ed.). "Sequence A084237". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Sloane, N. J. A. (ed.). "Sequence A084237". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Sloane, N. J. A. (ed.). "Sequence A084237". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Meissel, Ernst (1870). "Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen". Mathematische Annalen (in German). 2 (4): 636–642. doi:10.1007/BF01444045. ISSN 0025-5831. S2CID 119828499. <a href="https://eudml.org/doc/156468" target="_blank">https://eudml.org/doc/156468</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Lehmer, Derrick Henry (April 1, 1958). "ON THE EXACT NUMBER OF PRIMES LESS THAN A GIVEN LIMIT". Illinois J. Math. 3 (3): 381–388. Retrieved February 1, 2017. <a href="https://projecteuclid.org/download/pdf_1/euclid.ijm/1255455259" target="_blank">https://projecteuclid.org/download/pdf_1/euclid.ijm/1255455259</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Lagarias, Jeffrey; Miller, Victor; Odlyzko, Andrew (April 11, 1985). "Computing 
 
 
 
 π
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 x
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 {\displaystyle \pi (x)}
 
: The Meissel–Lehmer method" (PDF). Mathematics of Computation. 44 (170): 537–560. doi:10.1090/S0025-5718-1985-0777285-5. Retrieved September 13, 2016. <a href="https://www.ams.org/mcom/1985-44-170/S0025-5718-1985-0777285-5/S0025-5718-1985-0777285-5.pdf" target="_blank">https://www.ams.org/mcom/1985-44-170/S0025-5718-1985-0777285-5/S0025-5718-1985-0777285-5.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Rivat, Joöl; Deléglise, Marc (1996). "Computing the summation of the Möbius function". Experimental Mathematics. 5 (4): 291–295. doi:10.1080/10586458.1996.10504594. ISSN 1944-950X. S2CID 574146. <a href="https://projecteuclid.org/euclid.em/1047565447" target="_blank">https://projecteuclid.org/euclid.em/1047565447</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Helfgott, Harald; Thompson, Lola (2023). "Summing 
 
 
 
 μ
 (
 n
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 {\displaystyle \mu (n)}
 
: a faster elementary algorithm". Research in Number Theory. 9 (1): 6. doi:10.1007/s40993-022-00408-8. ISSN 2363-9555. PMC 9731940. PMID 36511765. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9731940" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9731940</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Lagarias, Jeffrey; Odlyzko, Andrew (June 1987). "Computing 
 
 
 
 π
 (
 x
 )
 
 
 {\displaystyle \pi (x)}
 
: An analytic method". Journal of Algorithms. 8 (2): 173–191. doi:10.1016/0196-6774(87)90037-X. <a href="https://www.sciencedirect.com/science/article/abs/pii/019667748790037X" target="_blank">https://www.sciencedirect.com/science/article/abs/pii/019667748790037X</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">El Marraki, M. (1995). "Fonction sommatoire de la fonction de Möbius, 3. Majorations asymptotiques effectives fortes". Journal de théorie des nombres de Bordeaux. 7 (2). <a href="http://www.numdam.org/item/JTNB_1995__7_2_407_0/" target="_blank">http://www.numdam.org/item/JTNB_1995__7_2_407_0/</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Mertens function open-in-new

Mertens function