Dirichlet L-function

<h2 id="euler-product">Euler product</h2>
Since a Dirichlet character 
 
 
 
 χ
 
 
 {\displaystyle \chi }
 
 is <a href="/facts/Completely_multiplicative/4Sr9ADXQ">completely multiplicative</a>, its 
 
 
 
 L
 
 
 {\displaystyle L}
 
-function can also be written as an <a href="/facts/Euler_product/Lvsv4U3M">Euler product</a> in the <a href="/facts/Half-plane/AGknBgnw">half-plane</a> of <a href="/facts/Absolute_convergence/BXz9kBJ5">absolute convergence</a>:

L
        (
        s
        ,
        χ
        )
        =
        
          ∏
          
            p
          
        
        
          
            (
            
              1
              −
              χ
              (
              p
              )
              
                p
                
                  −
                  s
                
              
            
            )
          
          
            −
            1
          
        
        
           for 
        
        
          Re
        
        (
        s
        )
        >
        1
        ,
      
    
    {\displaystyle L(s,\chi )=\prod _{p}\left(1-\chi (p)p^{-s}\right)^{-1}{\text{ for }}{\text{Re}}(s)>1,}

where the product is over all <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h2 id="primitive-characters">Primitive characters</h2>
Results about L-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> This is because of the relationship between a imprimitive character 
 
 
 
 χ
 
 
 {\displaystyle \chi }
 
 and the primitive character 
 
 
 
 
 χ
 
 ⋆
 
 
 
 
 {\displaystyle \chi ^{\star }}
 
 which induces it:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

χ
        (
        n
        )
        =
        
          
            {
            
              
                
                  
                    χ
                    
                      ⋆
                    
                  
                  (
                  n
                  )
                  ,
                
                
                  
                    i
                    f
                  
                  gcd
                  (
                  n
                  ,
                  q
                  )
                  =
                  1
                
              
              
                
                  0
                  ,
                
                
                  
                    i
                    f
                  
                  gcd
                  (
                  n
                  ,
                  q
                  )
                  ≠
                  1
                
              
            
            
          
        
      
    
    {\displaystyle \chi (n)={\begin{cases}\chi ^{\star }(n),&\mathrm {if} \gcd(n,q)=1\\0,&\mathrm {if} \gcd(n,q)\neq 1\end{cases}}}

(Here, q is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-functions:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

L
        (
        s
        ,
        χ
        )
        =
        L
        (
        s
        ,
        
          χ
          
            ⋆
          
        
        )
        
          ∏
          
            p
            
            
              |
            
            
            q
          
        
        
          (
          
            1
            −
            
              
                
                  
                    χ
                    
                      ⋆
                    
                  
                  (
                  p
                  )
                
                
                  p
                  
                    s
                  
                
              
            
          
          )
        
      
    
    {\displaystyle L(s,\chi )=L(s,\chi ^{\star })\prod _{p\,|\,q}\left(1-{\frac {\chi ^{\star }(p)}{p^{s}}}\right)}

(This formula holds for all s, by analytic continuation, even though the Euler product is only valid when Re(s) > 1.) The formula shows that the L-function of χ is equal to the L-function of the primitive character which induces χ, multiplied by only a finite number of factors.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
As a special case, the L-function of the principal character 
 
 
 
 
 χ
 
 0
 
 
 
 
 {\displaystyle \chi _{0}}
 
 modulo q can be expressed in terms of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

L
        (
        s
        ,
        
          χ
          
            0
          
        
        )
        =
        ζ
        (
        s
        )
        
          ∏
          
            p
            
            
              |
            
            
            q
          
        
        (
        1
        −
        
          p
          
            −
            s
          
        
        )
      
    
    {\displaystyle L(s,\chi _{0})=\zeta (s)\prod _{p\,|\,q}(1-p^{-s})}

<h2 id="functional-equation">Functional equation</h2>
Dirichlet L-functions satisfy a <a href="/facts/Functional_equation/4ovU4CAt">functional equation</a>, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of 
 
 
 
 L
 (
 s
 ,
 χ
 )
 
 
 {\displaystyle L(s,\chi )}
 
 to the value of 
 
 
 
 L
 (
 1
 −
 s
 ,
 
 
 χ
 ¯
 
 
 )
 
 
 {\displaystyle L(1-s,{\overline {\chi }})}
 
. Let χ be a primitive character modulo q, where q > 1. One way to express the functional equation is:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

L
        (
        s
        ,
        χ
        )
        =
        W
        (
        χ
        )
        
          2
          
            s
          
        
        
          π
          
            s
            −
            1
          
        
        
          q
          
            1
            
              /
            
            2
            −
            s
          
        
        sin
        ⁡
        
          (
          
            
              
                π
                2
              
            
            (
            s
            +
            δ
            )
          
          )
        
        Γ
        (
        1
        −
        s
        )
        L
        (
        1
        −
        s
        ,
        
          
            χ
            ¯
          
        
        )
        .
      
    
    {\displaystyle L(s,\chi )=W(\chi )2^{s}\pi ^{s-1}q^{1/2-s}\sin \left({\frac {\pi }{2}}(s+\delta )\right)\Gamma (1-s)L(1-s,{\overline {\chi }}).}

In this equation, Γ denotes the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>;

χ
        (
        −
        1
        )
        =
        (
        −
        1
        
          )
          
            δ
          
        
      
    
    {\displaystyle \chi (-1)=(-1)^{\delta }}
  
 ; and

W
        (
        χ
        )
        =
        
          
            
              τ
              (
              χ
              )
            
            
              
                i
                
                  δ
                
              
              
                
                  q
                
              
            
          
        
      
    
    {\displaystyle W(\chi )={\frac {\tau (\chi )}{i^{\delta }{\sqrt {q}}}}}

where τ ( χ) is a <a href="/facts/Gauss_sum/a87mpRfN">Gauss sum</a>:

τ
        (
        χ
        )
        =
        
          ∑
          
            a
            =
            1
          
          
            q
          
        
        χ
        (
        a
        )
        exp
        ⁡
        (
        2
        π
        i
        a
        
          /
        
        q
        )
        .
      
    
    {\displaystyle \tau (\chi )=\sum _{a=1}^{q}\chi (a)\exp(2\pi ia/q).}

It is a property of Gauss sums that |τ ( χ) | = q1/2, so |W ( χ) | = 1.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
Another way to state the functional equation is in terms of

Λ
        (
        s
        ,
        χ
        )
        =
        
          q
          
            s
            
              /
            
            2
          
        
        
          π
          
            −
            (
            s
            +
            δ
            )
            
              /
            
            2
          
        
        
          Γ
        
        ⁡
        
          (
          
            
              
                s
                +
                δ
              
              2
            
          
          )
        
        L
        (
        s
        ,
        χ
        )
        .
      
    
    {\displaystyle \Lambda (s,\chi )=q^{s/2}\pi ^{-(s+\delta )/2}\operatorname {\Gamma } \left({\frac {s+\delta }{2}}\right)L(s,\chi ).}

The functional equation can be expressed as:<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

Λ
        (
        s
        ,
        χ
        )
        =
        W
        (
        χ
        )
        Λ
        (
        1
        −
        s
        ,
        
          
            χ
            ¯
          
        
        )
        .
      
    
    {\displaystyle \Lambda (s,\chi )=W(\chi )\Lambda (1-s,{\overline {\chi }}).}

The functional equation implies that 
 
 
 
 L
 (
 s
 ,
 χ
 )
 
 
 {\displaystyle L(s,\chi )}
 
 (and 
 
 
 
 Λ
 (
 s
 ,
 χ
 )
 
 
 {\displaystyle \Lambda (s,\chi )}
 
) are <a href="/facts/Entire_function/XNluKzu7">entire functions</a> of s. (Again, this assumes that χ is primitive character modulo q with q > 1. If q = 1, then 
 
 
 
 L
 (
 s
 ,
 χ
 )
 =
 ζ
 (
 s
 )
 
 
 {\displaystyle L(s,\chi )=\zeta (s)}
 
 has a pole at s = 1.)<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
For generalizations, see: <a href="/facts/Functional_equation_(L-function)/qKa0wBC5">Functional equation (L-function)</a>.

<h2 id="zeros">Zeros</h2>

Let χ be a primitive character modulo q, with q > 1.
There are no <a href="/facts/Zero_of_a_function/mlK3dhoT">zeros</a> of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative <a href="/facts/Integer/PUwwrawx">integers</a> s:

<ul><li>If χ(−1) = 1, the only zeros of L(s, χ) with Re(s) < 0 are simple zeros at −2, −4, −6, .... (There is also a zero at s = 0.) These correspond to the poles of 
 
 
 
 
 Γ
 (
 
 
 s
 2
 
 
 )
 
 
 
 {\displaystyle \textstyle \Gamma ({\frac {s}{2}})}
 
.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li>
<li>If χ(−1) = −1, then the only zeros of L(s, χ) with Re(s) < 0 are simple zeros at −1, −3, −5, .... These correspond to the poles of 
 
 
 
 
 Γ
 (
 
 
 
 s
 +
 1
 
 2
 
 
 )
 
 
 
 {\displaystyle \textstyle \Gamma ({\frac {s+1}{2}})}
 
.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a></li></ul>
These are called the trivial zeros.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
The remaining zeros lie in the critical strip 0 ≤ Re(s) ≤ 1, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line Re(s) = 1/2. That is, if 
 
 
 
 L
 (
 ρ
 ,
 χ
 )
 =
 0
 
 
 {\displaystyle L(\rho ,\chi )=0}
 
 then 
 
 
 
 L
 (
 1
 −
 
 
 ρ
 ¯
 
 
 ,
 χ
 )
 =
 0
 
 
 {\displaystyle L(1-{\overline {\rho }},\chi )=0}
 
 too, because of the functional equation. If χ is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if χ is a complex character. The <a href="/facts/Generalized_Riemann_hypothesis/17nLmBbu">generalized Riemann hypothesis</a> is the conjecture that all the non-trivial zeros lie on the critical line Re(s) = 1/2.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
Up to the possible existence of a <a href="/facts/Siegel_zero/2sSmBoxW">Siegel zero</a>, zero-free regions including and beyond the line Re(s) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L-functions: for example, for χ a non-real character of modulus q, we have

β
 <
 1
 −
 
 
 c
 
 log
 
 
 
 
 
 (
 
 
 q
 (
 2
 +
 
 |
 
 γ
 
 |
 
 )
 
 
 )
 
 
 
 
 
  
 
 
 {\displaystyle \beta <1-{\frac {c}{\log \!\!\;{\big (}q(2+|\gamma |){\big )}}}\ }

for β + iγ a non-real zero.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>

<h2 id="relation-to-the-hurwitz-zeta-function">Relation to the Hurwitz zeta function</h2>
The Dirichlet L-functions may be written as a linear combination of the <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a> at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,a) where a = r/k and r = 1, 2, ..., k. This means that the Hurwitz zeta function for rational a has analytic properties that are closely related to the Dirichlet L-functions. Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as:<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

L
        (
        s
        ,
        χ
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              χ
              (
              n
              )
            
            
              n
              
                s
              
            
          
        
        =
        
          
            1
            
              k
              
                s
              
            
          
        
        
          ∑
          
            r
            =
            1
          
          
            k
          
        
        χ
        (
        r
        )
        
          ζ
        
        ⁡
        
          (
          
            s
            ,
            
              
                r
                k
              
            
          
          )
        
        .
      
    
    {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}={\frac {1}{k^{s}}}\sum _{r=1}^{k}\chi (r)\operatorname {\zeta } \left(s,{\frac {r}{k}}\right).}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Generalized_Riemann_hypothesis/17nLmBbu">Generalized Riemann hypothesis</a></li>
<li><a href="/facts/L-function/REVCKvU7">L-function</a></li>
<li><a href="/facts/Modularity_theorem/aI3HRxwj">Modularity theorem</a></li>
<li><a href="/facts/Artin_conjecture_(L-functions)/XeNKVPPR">Artin conjecture</a></li>
<li><a href="/facts/Special_values_of_L-functions/enVGIRfP">Special values of L-functions</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><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>Apostol, T. M. (2010), <a href="http://dlmf.nist.gov/25.15">"Dirichlet L-function"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li>
<li><a href="/facts/Harold_Davenport/aceBKgHd">Davenport, H.</a> (2000). Multiplicative Number Theory (3rd ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95097-4.</li>
<li><a href="/facts/Peter_Gustav_Lejeune_Dirichlet/b5UIU7Uy">Dirichlet, P. G. L.</a> (1837). "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält". Abhand. Ak. Wiss. Berlin. 48.</li>
<li>Ireland, Kenneth; <a href="/facts/Michael_Rosen_(mathematician)/63pUD7gC">Rosen, Michael</a> (1990). A Classical Introduction to Modern Number Theory (2nd ed.). Springer-Verlag.</li>
<li><a href="/facts/Hugh_Montgomery_(mathematician)/k1VI3W69">Montgomery, Hugh L.</a>; <a href="/facts/Robert_Charles_Vaughan_(mathematician)/djBcTCVL">Vaughan, Robert C.</a> (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-84903-6.</li>
<li><a href="/facts/Henryk_Iwaniec/4yVGj3lU">Iwaniec, Henryk</a>; Kowalski, Emmanuel (2004). Analytic Number Theory. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society.</li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=Dirichlet-L-function">"Dirichlet-L-function"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

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

<ol>
<li id="fn:1">Apostol 1976, Theorem 11.7 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0434929</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Davenport 2000, chapter 5 - Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer. ISBN 0-387-95097-4. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Davenport 2000, chapter 5, equation (2) - Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer. ISBN 0-387-95097-4. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Davenport 2000, chapter 5, equation (3) - Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer. ISBN 0-387-95097-4. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Montgomery & Vaughan 2006, p. 282 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Apostol 1976, p. 262 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0434929</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Ireland & Rosen 1990, chapter 16, section 4 - Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory (2nd ed.). Springer-Verlag. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Montgomery & Vaughan 2006, p. 121 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Montgomery & Vaughan 2006, p. 333 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Montgomery & Vaughan 2006, p. 332 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Iwaniec & Kowalski 2004, p. 84 - Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic Number Theory. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Montgomery & Vaughan 2006, p. 333 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Iwaniec & Kowalski 2004, p. 84 - Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic Number Theory. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Montgomery & Vaughan 2006, p. 333 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Iwaniec & Kowalski 2004, p. 84 - Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic Number Theory. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Davenport 2000, chapter 9 - Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer. ISBN 0-387-95097-4. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Davenport 2000, chapter 9 - Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer. ISBN 0-387-95097-4. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Montgomery & Vaughan 2006, p. 333 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Montgomery & Vaughan 2006, p. 333 - Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. p. 163. ISBN 0-8218-0737-4. Zbl 0814.11001. <a href="0-8218-0737-4" target="_blank">0-8218-0737-4</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Apostol 1976, p. 249 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0434929</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
</ol>

Dirichlet L-function open-in-new

Dirichlet L-function