Perron's formula

<h2 id="statement">Statement</h2>
Let 
 
 
 
 {
 a
 (
 n
 )
 }
 
 
 {\displaystyle \{a(n)\}}
 
 be an <a href="/facts/Arithmetic_function/hULxwsgm">arithmetic function</a>, and let

g
        (
        s
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              a
              (
              n
              )
            
            
              n
              
                s
              
            
          
        
      
    
    {\displaystyle g(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}}

be the corresponding <a href="/facts/Dirichlet_series/uIAUaLcd">Dirichlet series</a>. Presume the Dirichlet series to be <a href="/facts/Uniform_convergence/ecz9eNWn">uniformly convergent</a> for 
 
 
 
 ℜ
 (
 s
 )
 >
 σ
 
 
 {\displaystyle \Re (s)>\sigma }
 
. Then Perron's formula is

A
        (
        x
        )
        =
        
          
            
              ∑
              
                n
                ≤
                x
              
            
          
          ′
        
        a
        (
        n
        )
        =
        
          
            1
            
              2
              π
              i
            
          
        
        
          ∫
          
            c
            −
            i
            ∞
          
          
            c
            +
            i
            ∞
          
        
        g
        (
        z
        )
        
          
            
              x
              
                z
              
            
            z
          
        
        
        d
        z
        .
      
    
    {\displaystyle A(x)={\sum _{n\leq x}}'a(n)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }g(z){\frac {x^{z}}{z}}\,dz.}

Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when x is an <a href="/facts/Integer/PUwwrawx">integer</a>. The integral is not a convergent <a href="/facts/Lebesgue_integral/f4AgvC6x">Lebesgue integral</a>; it is understood as the <a href="/facts/Cauchy_principal_value/Gm33Qevf">Cauchy principal value</a>. The formula requires that c > 0, c > σ, and x > 0.

<h2 id="proof">Proof</h2>
An easy sketch of the proof comes from taking <a href="/facts/Abel%27s_summation_formula/pgD6MYXg">Abel's sum formula</a>

g
        (
        s
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              a
              (
              n
              )
            
            
              n
              
                s
              
            
          
        
        =
        s
        
          ∫
          
            1
          
          
            ∞
          
        
        A
        (
        x
        )
        
          x
          
            −
            (
            s
            +
            1
            )
          
        
        d
        x
        .
      
    
    {\displaystyle g(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}=s\int _{1}^{\infty }A(x)x^{-(s+1)}dx.}

This is nothing but a <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> under the variable change 
 
 
 
 x
 =
 
 e
 
 t
 
 
 .
 
 
 {\displaystyle x=e^{t}.}
 
 Inverting it one gets Perron's formula.

<h2 id="examples">Examples</h2>
Because of its general relationship to Dirichlet series, the formula is commonly applied to many number-theoretic sums. Thus, for example, one has the famous integral representation for the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>:

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

and a similar formula for <a href="/facts/Dirichlet_L-function/my8hc6Qy">Dirichlet L-functions</a>:

L
        (
        s
        ,
        χ
        )
        =
        s
        
          ∫
          
            1
          
          
            ∞
          
        
        
          
            
              A
              (
              x
              )
            
            
              x
              
                s
                +
                1
              
            
          
        
        
        d
        x
      
    
    {\displaystyle L(s,\chi )=s\int _{1}^{\infty }{\frac {A(x)}{x^{s+1}}}\,dx}

where

A
        (
        x
        )
        =
        
          ∑
          
            n
            ≤
            x
          
        
        χ
        (
        n
        )
      
    
    {\displaystyle A(x)=\sum _{n\leq x}\chi (n)}

and 
 
 
 
 χ
 (
 n
 )
 
 
 {\displaystyle \chi (n)}
 
 is a <a href="/facts/Dirichlet_character/jDJHEBWa">Dirichlet character</a>. Other examples appear in the articles on the <a href="/facts/Mertens_function/ardK1BJv">Mertens function</a> and the <a href="/facts/Von_Mangoldt_function/wwHKs2bg">von Mangoldt function</a>.

<h2 id="generalizations">Generalizations</h2>
Perron's formula is just a special case of the formula

∑
          
            n
            =
            1
          
          
            ∞
          
        
        a
        (
        n
        )
        f
        (
        n
        
          /
        
        x
        )
        =
        
          
            1
            
              2
              π
              i
            
          
        
        
          ∫
          
            c
            −
            i
            ∞
          
          
            c
            +
            i
            ∞
          
        
        F
        (
        s
        )
        G
        (
        s
        )
        
          x
          
            s
          
        
        d
        s
      
    
    {\displaystyle \sum _{n=1}^{\infty }a(n)f(n/x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }F(s)G(s)x^{s}ds}

where

G
        (
        s
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              a
              (
              n
              )
            
            
              n
              
                s
              
            
          
        
      
    
    {\displaystyle G(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}}

and

F
        (
        s
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        f
        (
        x
        )
        
          x
          
            s
            −
            1
          
        
        d
        x
      
    
    {\displaystyle F(s)=\int _{0}^{\infty }f(x)x^{s-1}dx}

the Mellin transform. The Perron formula is just the special case of the test function 
 
 
 
 f
 (
 1
 
 /
 
 x
 )
 =
 θ
 (
 x
 −
 1
 )
 ,
 
 
 {\displaystyle f(1/x)=\theta (x-1),}
 
 for 
 
 
 
 θ
 (
 x
 )
 
 
 {\displaystyle \theta (x)}
 
 the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>.

<ul><li>Page 243 of <a href="/facts/Tom_M._Apostol/S6vGtJWN">Apostol, Tom M.</a> (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90163-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/PerronsFormula.html">"Perron's formula"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li>Tenenbaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics. Vol. 46. Translated by C.B. Thomas. Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-41261-7. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0831.11001">0831.11001</a>.</li></ul>

Perron's formula open-in-new

Perron's formula