In mathematics , a Dirichlet L -series is a function of the form
92-499: Where χ {\displaystyle \chi } is a Dirichlet character and s a complex variable with real part greater than 1. It is a special case of a Dirichlet series . By analytic continuation , it can be extended to a meromorphic function on the whole complex plane , and is then called a Dirichlet L -function and also denoted L ( s , χ ). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in ( Dirichlet 1837 ) to prove
184-779: A , a ∈ ( 0 , 1 ) . {\displaystyle I=\int _{-\infty }^{\infty }{\frac {e^{ax}}{1+e^{x}}}\,dx=\int _{0}^{\infty }{\frac {v^{a-1}}{1+v}}\,dv={\frac {\pi }{\sin \pi a}},\quad a\in (0,1).} Consider the positively oriented rectangular contour C R {\displaystyle C_{R}} with vertices at R {\displaystyle R} , − R {\displaystyle -R} , R + 2 π i {\displaystyle R+2\pi i} and − R + 2 π i {\displaystyle -R+2\pi i} where R ∈ R + {\displaystyle R\in \mathbb {R} ^{+}} . Then by
276-419: A I R {\displaystyle I_{R}'=-e^{2\pi ia}I_{R}} . If A R {\displaystyle A_{R}} denotes the right vertical side of the rectangle, then | ∫ A R e a z 1 + e z d z | ≤ ∫ 0 2 π | e
368-1860: A I = − 2 π i e a π i , {\displaystyle I-e^{2\pi ia}I=-2\pi ie^{a\pi i},} from which I = π sin π a , a ∈ ( 0 , 1 ) . {\displaystyle I={\frac {\pi }{\sin \pi a}},\quad a\in (0,1).} Then Γ ( 1 − z ) = ∫ 0 ∞ e − u u − z d u = t ∫ 0 ∞ e − v t ( v t ) − z d v , t > 0 {\displaystyle \Gamma (1-z)=\int _{0}^{\infty }e^{-u}u^{-z}\,du=t\int _{0}^{\infty }e^{-vt}(vt)^{-z}\,dv,\quad t>0} and Γ ( z ) Γ ( 1 − z ) = ∫ 0 ∞ ∫ 0 ∞ e − t ( 1 + v ) v − z d v d t = ∫ 0 ∞ v − z 1 + v d v = π sin π ( 1 − z ) = π sin π z , z ∈ ( 0 , 1 ) . {\displaystyle {\begin{aligned}\Gamma (z)\Gamma (1-z)&=\int _{0}^{\infty }\int _{0}^{\infty }e^{-t(1+v)}v^{-z}\,dv\,dt\\&=\int _{0}^{\infty }{\frac {v^{-z}}{1+v}}\,dv\\&={\frac {\pi }{\sin \pi (1-z)}}\\&={\frac {\pi }{\sin \pi z}},\quad z\in (0,1).\end{aligned}}} Proving
460-555: A {\displaystyle a} ) by For ( a b , q ) = 1 , a ≡ b ( mod q ) {\displaystyle (ab,q)=1,\;\;a\equiv b{\pmod {q}}} if and only if ν q ( a ) = ν q ( b ) . {\displaystyle \nu _{q}(a)=\nu _{q}(b).} Since Let ω q = ζ ϕ ( q ) {\displaystyle \omega _{q}=\zeta _{\phi (q)}} be
552-432: A ) − 1 {\displaystyle \eta ^{-1}(a)=\eta (a)^{-1}} then G ^ {\displaystyle {\widehat {G}}} becomes an abelian group. If A {\displaystyle A} is a finite abelian group then there is an isomorphism A ≅ A ^ {\displaystyle A\cong {\widehat {A}}} , and
644-488: A < 1 {\displaystyle a<1} , the integral tends to 0 {\displaystyle 0} as R → ∞ {\displaystyle R\to \infty } . Analogously, the integral over the left vertical side of the rectangle tends to 0 {\displaystyle 0} as R → ∞ {\displaystyle R\to \infty } . Therefore I − e 2 π i
736-423: A ( R + i t ) 1 + e R + i t | d t ≤ C e ( a − 1 ) R {\displaystyle \left|\int _{A_{R}}{\frac {e^{az}}{1+e^{z}}}\,dz\right|\leq \int _{0}^{2\pi }\left|{\frac {e^{a(R+it)}}{1+e^{R+it}}}\right|\,dt\leq Ce^{(a-1)R}} for some constant C {\displaystyle C} and since
828-507: A ) {\displaystyle \chi _{q,r}(a)} as Then for ( r s , q ) = 1 {\displaystyle (rs,q)=1} and all a {\displaystyle a} and b {\displaystyle b} 2 is a primitive root mod 3. ( ϕ ( 3 ) = 2 {\displaystyle \phi (3)=2} ) so the values of ν 3 {\displaystyle \nu _{3}} are The nonzero values of
920-399: A ) , χ ′ ( a ) , χ r ( a ) , {\displaystyle \chi (a),\chi '(a),\chi _{r}(a),} etc. are Dirichlet characters. (the lowercase Greek letter chi for "character") There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem)
1012-405: A ) = η ( a ) θ ( a ) , {\displaystyle \eta \theta (a)=\eta (a)\theta (a),} the identity by the trivial character η 0 ( a ) = 1 {\displaystyle \eta _{0}(a)=1} and the inverse by complex inversion η − 1 ( a ) = η (
SECTION 10
#17327810120611104-587: A , m ) = 1 {\displaystyle (a,m)=1} Thus for all integers a {\displaystyle a} 10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a finite abelian group . There are three different cases because the groups ( Z / m Z ) × {\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }} have different structures depending on whether m {\displaystyle m}
1196-531: A , m ) = 1. {\displaystyle (a,m)=1.} A group character ρ : ( Z / m Z ) × → C × {\displaystyle \rho :(\mathbb {Z} /m\mathbb {Z} )^{\times }\rightarrow \mathbb {C} ^{\times }} can be extended to a Dirichlet character χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } by defining and conversely,
1288-554: A x 1 + e x d x {\displaystyle I_{R}=\int _{-R}^{R}{\frac {e^{ax}}{1+e^{x}}}\,dx} and let I R ′ {\displaystyle I_{R}'} be the analogous integral over the top side of the rectangle. Then I R → I {\displaystyle I_{R}\to I} as R → ∞ {\displaystyle R\to \infty } and I R ′ = − e 2 π i
1380-456: A + bi , this product is | Γ ( a + b i ) | 2 = | Γ ( a ) | 2 ∏ k = 0 ∞ 1 1 + b 2 ( a + k ) 2 {\displaystyle |\Gamma (a+bi)|^{2}=|\Gamma (a)|^{2}\prod _{k=0}^{\infty }{\frac {1}{1+{\frac {b^{2}}{(a+k)^{2}}}}}} If
1472-1388: A Gaussian integral . In general, for non-negative integer values of n {\displaystyle n} we have: Γ ( 1 2 + n ) = ( 2 n ) ! 4 n n ! π = ( 2 n − 1 ) ! ! 2 n π = ( n − 1 2 n ) n ! π Γ ( 1 2 − n ) = ( − 4 ) n n ! ( 2 n ) ! π = ( − 2 ) n ( 2 n − 1 ) ! ! π = π ( − 1 / 2 n ) n ! {\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \over 4^{n}n!}{\sqrt {\pi }}={\frac {(2n-1)!!}{2^{n}}}{\sqrt {\pi }}={\binom {n-{\frac {1}{2}}}{n}}n!{\sqrt {\pi }}\\[8pt]\Gamma \left({\tfrac {1}{2}}-n\right)&={(-4)^{n}n! \over (2n)!}{\sqrt {\pi }}={\frac {(-2)^{n}}{(2n-1)!!}}{\sqrt {\pi }}={\frac {\sqrt {\pi }}{{\binom {-1/2}{n}}n!}}\end{aligned}}} where
1564-406: A Dirichlet character mod m {\displaystyle m} defines a group character on ( Z / m Z ) × . {\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }.} Paraphrasing Davenport Dirichlet characters can be regarded as a particular case of Abelian group characters. But this article follows Dirichlet in giving
1656-490: A complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character of modulus m {\displaystyle m} (where m {\displaystyle m} is a positive integer) if for all integers a {\displaystyle a} and b {\displaystyle b} : The simplest possible character, called
1748-450: A convergent improper integral for complex numbers with positive real part: Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t , ℜ ( z ) > 0 . {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}{\text{ d}}t,\ \qquad \Re (z)>0\,.} The gamma function then
1840-864: A direct and constructive account of them. This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group. 4) Since gcd ( 1 , m ) = 1 , {\displaystyle \gcd(1,m)=1,} property 2) says χ ( 1 ) ≠ 0 {\displaystyle \chi (1)\neq 0} so it can be canceled from both sides of χ ( 1 ) χ ( 1 ) = χ ( 1 × 1 ) = χ ( 1 ) {\displaystyle \chi (1)\chi (1)=\chi (1\times 1)=\chi (1)} : 5) Property 3)
1932-417: A finite number of characters for a given modulus. 8) If χ {\displaystyle \chi } and χ ′ {\displaystyle \chi '} are two characters for the same modulus so is their product χ χ ′ , {\displaystyle \chi \chi ',} defined by pointwise multiplication: The principal character
SECTION 20
#17327810120612024-465: A fixed integer m {\displaystyle m} , as the integer n {\displaystyle n} increases, we have that lim n → ∞ n ! ( n + 1 ) m ( n + m ) ! = 1 . {\displaystyle \lim _{n\to \infty }{\frac {n!\,\left(n+1\right)^{m}}{(n+m)!}}=1\,.} If m {\displaystyle m}
2116-618: A formula for other individual values Γ ( r ) {\displaystyle \Gamma (r)} where r {\displaystyle r} is rational, especially because according to Gauss's digamma theorem , it is possible to do so for the closely related digamma function at every rational value. However, these numbers Γ ( r ) {\displaystyle \Gamma (r)} are not known to be expressible by themselves in terms of elementary functions. It has been proved that Γ ( n + r ) {\displaystyle \Gamma (n+r)}
2208-398: A generator is called a primitive root mod q {\displaystyle q} . Let g q {\displaystyle g_{q}} be a primitive root and for ( a , q ) = 1 {\displaystyle (a,q)=1} define the function ν q ( a ) {\displaystyle \nu _{q}(a)} (the index of
2300-829: A primitive ϕ ( q ) {\displaystyle \phi (q)} -th root of unity. From property 7) above the possible values of χ ( g q ) {\displaystyle \chi (g_{q})} are ω q , ω q 2 , . . . ω q ϕ ( q ) = 1. {\displaystyle \omega _{q},\omega _{q}^{2},...\omega _{q}^{\phi (q)}=1.} These distinct values give rise to ϕ ( q ) {\displaystyle \phi (q)} Dirichlet characters mod q . {\displaystyle q.} For ( r , q ) = 1 {\displaystyle (r,q)=1} define χ q , r (
2392-524: A unique solution, since it allows for multiplication by any periodic function g ( x ) {\displaystyle g(x)} with g ( x ) = g ( x + 1 ) {\displaystyle g(x)=g(x+1)} and g ( 0 ) = 1 {\displaystyle g(0)=1} , such as g ( x ) = e k sin ( m π x ) {\displaystyle g(x)=e^{k\sin(m\pi x)}} . One way to resolve
2484-440: 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
2576-643: Is Euler's totient function . ζ n {\displaystyle \zeta _{n}} is a complex primitive n-th root of unity : ( Z / m Z ) × {\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }} is the group of units mod m {\displaystyle m} . It has order ϕ ( m ) . {\displaystyle \phi (m).} ( Z / m Z ) × ^ {\displaystyle {\widehat {(\mathbb {Z} /m\mathbb {Z} )^{\times }}}}
2668-583: Is convex . The notation Γ ( z ) {\displaystyle \Gamma (z)} is due to Legendre . If the real part of the complex number z is strictly positive ( ℜ ( z ) > 0 {\displaystyle \Re (z)>0} ), then the integral Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt} converges absolutely , and
2760-518: Is a transcendental number and algebraically independent of π {\displaystyle \pi } for any integer n {\displaystyle n} and each of the fractions r = 1 6 , 1 4 , 1 3 , 2 3 , 3 4 , 5 6 {\textstyle r={\frac {1}{6}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{6}}} . In general, when computing values of
2852-408: Is a power of 2, a power of an odd prime, or the product of prime powers. If q = p k {\displaystyle q=p^{k}} is an odd number ( Z / q Z ) × {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }} is cyclic of order ϕ ( q ) {\displaystyle \phi (q)} ;
Dirichlet L-function - Misplaced Pages Continue
2944-736: Is an entire function , converging for every complex number z . The definition for the gamma function due to Weierstrass is also valid for all complex numbers z {\displaystyle z} except non-positive integers: Γ ( z ) = e − γ z z ∏ n = 1 ∞ ( 1 + z n ) − 1 e z / n , {\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n},} where γ ≈ 0.577216 {\displaystyle \gamma \approx 0.577216}
3036-3559: Is an entire function of order 1 {\displaystyle 1} . Since z Γ ( z ) → 1 {\displaystyle z\Gamma (z)\to 1} as z → 0 {\displaystyle z\to 0} , c 2 = 0 {\displaystyle c_{2}=0} (or an integer multiple of 2 π i {\displaystyle 2\pi i} ) and since Γ ( 1 ) = 1 {\displaystyle \Gamma (1)=1} , e − c 1 = ∏ n = 1 ∞ e − 1 n ( 1 + 1 n ) = exp ( lim N → ∞ ∑ n = 1 N ( log ( 1 + 1 n ) − 1 n ) ) = exp ( lim N → ∞ ( log ( N + 1 ) − ∑ n = 1 N 1 n ) ) = exp ( lim N → ∞ ( log N + log ( 1 + 1 N ) − ∑ n = 1 N 1 n ) ) = exp ( lim N → ∞ ( log N − ∑ n = 1 N 1 n ) ) = e − γ . {\displaystyle {\begin{aligned}e^{-c_{1}}&=\prod _{n=1}^{\infty }e^{-{\frac {1}{n}}}\left(1+{\frac {1}{n}}\right)\\&=\exp \left(\lim _{N\to \infty }\sum _{n=1}^{N}\left(\log \left(1+{\frac {1}{n}}\right)-{\frac {1}{n}}\right)\right)\\&=\exp \left(\lim _{N\to \infty }\left(\log(N+1)-\sum _{n=1}^{N}{\frac {1}{n}}\right)\right)\\&=\exp \left(\lim _{N\to \infty }\left(\log N+\log \left(1+{\frac {1}{N}}\right)-\sum _{n=1}^{N}{\frac {1}{n}}\right)\right)\\&=\exp \left(\lim _{N\to \infty }\left(\log N-\sum _{n=1}^{N}{\frac {1}{n}}\right)\right)\\&=e^{-\gamma }.\end{aligned}}} where c 1 = γ + 2 π i k {\displaystyle c_{1}=\gamma +2\pi ik} for some integer k {\displaystyle k} . Since Γ ( z ) ∈ R {\displaystyle \Gamma (z)\in \mathbb {R} } for z ∈ R ∖ Z 0 − {\displaystyle z\in \mathbb {R} \setminus \mathbb {Z} _{0}^{-}} , we have k = 0 {\displaystyle k=0} and 1 Γ ( z ) = z e γ z ∏ n = 1 ∞ e − z n ( 1 + z n ) , z ∈ C ∖ Z 0 − . {\displaystyle {\frac {1}{\Gamma (z)}}=ze^{\gamma z}\prod _{n=1}^{\infty }e^{-{\frac {z}{n}}}\left(1+{\frac {z}{n}}\right),\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}.} Equivalence of
3128-431: Is an identity: 9) Let a − 1 {\displaystyle a^{-1}} denote the inverse of a {\displaystyle a} in ( Z / m Z ) × {\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }} . Then The complex conjugate of a root of unity is also its inverse (see here for details), so for (
3220-448: Is approximately the result of computing Γ ( n + 1 ) = n ! {\displaystyle \Gamma (n+1)=n!} for some large integer n {\displaystyle n} , multiplying by ( n + 1 ) z {\displaystyle (n+1)^{z}} to approximate Γ ( n + z + 1 ) {\displaystyle \Gamma (n+z+1)} , and using
3312-463: Is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles . The gamma function has no zeros, so the reciprocal gamma function 1 / Γ( z ) is an entire function . In fact, the gamma function corresponds to the Mellin transform of
3404-398: Is due to Euler, converges for all complex numbers z {\displaystyle z} except the non-positive integers, which fail because of a division by zero. Hence the above assumption produces a unique definition of z ! {\displaystyle z!} . Intuitively, this formula indicates that Γ ( z ) {\displaystyle \Gamma (z)}
3496-431: Is equivalent to 6) Property 1) implies that, for any positive integer n {\displaystyle n} 7) Euler's theorem states that if ( a , m ) = 1 {\displaystyle (a,m)=1} then a ϕ ( m ) ≡ 1 ( mod m ) . {\displaystyle a^{\phi (m)}\equiv 1{\pmod {m}}.} Therefore, That is,
3588-2107: Is even, hence 2 2 z − 1 Γ 2 ( z ) = 2 Γ ( 2 z ) ∫ 0 1 ( 1 − u 2 ) z − 1 d u . {\displaystyle 2^{2z-1}\Gamma ^{2}(z)=2\Gamma (2z)\int _{0}^{1}(1-u^{2})^{z-1}\,du.} Now assume B ( 1 2 , z ) = ∫ 0 1 t 1 2 − 1 ( 1 − t ) z − 1 d t , t = s 2 . {\displaystyle \mathrm {B} \left({\frac {1}{2}},z\right)=\int _{0}^{1}t^{{\frac {1}{2}}-1}(1-t)^{z-1}\,dt,\quad t=s^{2}.} Then B ( 1 2 , z ) = 2 ∫ 0 1 ( 1 − s 2 ) z − 1 d s = 2 ∫ 0 1 ( 1 − u 2 ) z − 1 d u . {\displaystyle \mathrm {B} \left({\frac {1}{2}},z\right)=2\int _{0}^{1}(1-s^{2})^{z-1}\,ds=2\int _{0}^{1}(1-u^{2})^{z-1}\,du.} This implies 2 2 z − 1 Γ 2 ( z ) = Γ ( 2 z ) B ( 1 2 , z ) . {\displaystyle 2^{2z-1}\Gamma ^{2}(z)=\Gamma (2z)\mathrm {B} \left({\frac {1}{2}},z\right).} Since B ( 1 2 , z ) = Γ ( 1 2 ) Γ ( z ) Γ ( z + 1 2 ) , Γ ( 1 2 ) = π , {\displaystyle \mathrm {B} \left({\frac {1}{2}},z\right)={\frac {\Gamma \left({\frac {1}{2}}\right)\Gamma (z)}{\Gamma \left(z+{\frac {1}{2}}\right)}},\quad \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }},}
3680-2797: Is known as the Euler integral of the second kind . (Euler's integral of the first kind is the beta function . ) Using integration by parts , one sees that: Γ ( z + 1 ) = ∫ 0 ∞ t z e − t d t = [ − t z e − t ] 0 ∞ + ∫ 0 ∞ z t z − 1 e − t d t = lim t → ∞ ( − t z e − t ) − ( − 0 z e − 0 ) + z ∫ 0 ∞ t z − 1 e − t d t . {\displaystyle {\begin{aligned}\Gamma (z+1)&=\int _{0}^{\infty }t^{z}e^{-t}\,dt\\&={\Bigl [}-t^{z}e^{-t}{\Bigr ]}_{0}^{\infty }+\int _{0}^{\infty }zt^{z-1}e^{-t}\,dt\\&=\lim _{t\to \infty }\left(-t^{z}e^{-t}\right)-\left(-0^{z}e^{-0}\right)+z\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.\end{aligned}}} Recognizing that − t z e − t → 0 {\displaystyle -t^{z}e^{-t}\to 0} as t → ∞ , {\displaystyle t\to \infty ,} Γ ( z + 1 ) = z ∫ 0 ∞ t z − 1 e − t d t = z Γ ( z ) . {\displaystyle {\begin{aligned}\Gamma (z+1)&=z\int _{0}^{\infty }t^{z-1}e^{-t}\,dt\\&=z\Gamma (z).\end{aligned}}} Then Γ ( 1 ) {\displaystyle \Gamma (1)} can be calculated as: Γ ( 1 ) = ∫ 0 ∞ t 1 − 1 e − t d t = ∫ 0 ∞ e − t d t = 1. {\displaystyle {\begin{aligned}\Gamma (1)&=\int _{0}^{\infty }t^{1-1}e^{-t}\,dt\\&=\int _{0}^{\infty }e^{-t}\,dt\\&=1.\end{aligned}}} Thus we can show that Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!} for any positive integer n by induction . Specifically,
3772-2164: Is not an integer, then this equation is meaningless, since in this section the factorial of a non-integer has not been defined yet. However, let us assume that this equation continues to hold when m {\displaystyle m} is replaced by an arbitrary complex number z {\displaystyle z} , in order to define the Gamma function for non integers: lim n → ∞ n ! ( n + 1 ) z ( n + z ) ! = 1 . {\displaystyle \lim _{n\to \infty }{\frac {n!\,\left(n+1\right)^{z}}{(n+z)!}}=1\,.} Multiplying both sides by ( z − 1 ) ! {\displaystyle (z-1)!} gives Γ ( z ) = ( z − 1 ) ! = 1 z lim n → ∞ n ! z ! ( n + z ) ! ( n + 1 ) z = 1 z lim n → ∞ ( 1 ⋅ 2 ⋯ n ) 1 ( 1 + z ) ⋯ ( n + z ) ( 2 1 ⋅ 3 2 ⋯ n + 1 n ) z = 1 z ∏ n = 1 ∞ [ 1 1 + z n ( 1 + 1 n ) z ] . {\displaystyle {\begin{aligned}\Gamma (z)&=(z-1)!\\[8pt]&={\frac {1}{z}}\lim _{n\to \infty }n!{\frac {z!}{(n+z)!}}(n+1)^{z}\\[8pt]&={\frac {1}{z}}\lim _{n\to \infty }(1\cdot 2\cdots n){\frac {1}{(1+z)\cdots (n+z)}}\left({\frac {2}{1}}\cdot {\frac {3}{2}}\cdots {\frac {n+1}{n}}\right)^{z}\\[8pt]&={\frac {1}{z}}\prod _{n=1}^{\infty }\left[{\frac {1}{1+{\frac {z}{n}}}}\left(1+{\frac {1}{n}}\right)^{z}\right].\end{aligned}}} This infinite product , which
Dirichlet L-function - Misplaced Pages Continue
3864-509: Is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as k sin ( m π x ) {\displaystyle k\sin(m\pi x)} for an integer m {\displaystyle m} . Such a function is known as a pseudogamma function , the most famous being the Hadamard function. A more restrictive requirement
3956-502: 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.) For generalizations, see: Functional equation (L-function) . Let χ be a primitive character modulo q , with q > 1. There are no zeros of L ( s , χ ) with Re( s ) > 1. For Re( s ) < 0, there are zeros at certain negative integers s : These are called
4048-1330: Is the Euler–Mascheroni constant . This is the Hadamard product of 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} in a rewritten form. This definition appears in an important identity involving pi. Equivalence of the integral definition and Weierstrass definition By the integral definition, the relation Γ ( z + 1 ) = z Γ ( z ) {\displaystyle \Gamma (z+1)=z\Gamma (z)} and Hadamard factorization theorem , 1 Γ ( z ) = z e c 1 z + c 2 ∏ n = 1 ∞ e − z n ( 1 + z n ) , z ∈ C ∖ Z 0 − {\displaystyle {\frac {1}{\Gamma (z)}}=ze^{c_{1}z+c_{2}}\prod _{n=1}^{\infty }e^{-{\frac {z}{n}}}\left(1+{\frac {z}{n}}\right),\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}} for some constants c 1 , c 2 {\displaystyle c_{1},c_{2}} since 1 / Γ {\displaystyle 1/\Gamma }
4140-474: Is the functional equation which interpolates the shifted factorial f ( n ) = ( n − 1 ) ! {\displaystyle f(n)=(n{-}1)!} : f ( x + 1 ) = x f ( x ) for any x > 0 , f ( 1 ) = 1. {\displaystyle f(x+1)=xf(x)\ {\text{ for any }}x>0,\qquad f(1)=1.} But this still does not give
4232-505: Is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order ϕ ( q ) 2 {\displaystyle {\frac {\phi (q)}{2}}} (generated by 5). For odd numbers a {\displaystyle a} define the functions ν 0 {\displaystyle \nu _{0}} and ν q {\displaystyle \nu _{q}} by Gamma function In mathematics ,
4324-411: Is the group of Dirichlet characters mod m {\displaystyle m} . p , p k , {\displaystyle p,p_{k},} etc. are prime numbers . ( m , n ) {\displaystyle (m,n)} is a standard abbreviation for gcd ( m , n ) {\displaystyle \gcd(m,n)} χ (
4416-425: Is the trivial group with one element. ( Z / 4 Z ) × {\displaystyle (\mathbb {Z} /4\mathbb {Z} )^{\times }} is cyclic of order 2. For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units ≡ 1 ( mod 4 ) {\displaystyle \equiv 1{\pmod {4}}} and their negatives are
4508-465: Is used several ways in mathematics. In this section it refers to a homomorphism from a group G {\displaystyle G} (written multiplicatively) to the multiplicative group of the field of complex numbers: The set of characters is denoted G ^ . {\displaystyle {\widehat {G}}.} If the product of two characters is defined by pointwise multiplication η θ (
4600-482: 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. 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 Riemann zeta function : Dirichlet L -functions satisfy a functional equation , which provides
4692-1774: The Legendre duplication formula Γ ( z ) Γ ( z + 1 2 ) = 2 1 − 2 z π Γ ( 2 z ) . {\displaystyle \Gamma (z)\Gamma \left(z+{\tfrac {1}{2}}\right)=2^{1-2z}\;{\sqrt {\pi }}\;\Gamma (2z).} Proof 1 With Euler's infinite product Γ ( z ) = 1 z ∏ n = 1 ∞ ( 1 + 1 / n ) z 1 + z / n {\displaystyle \Gamma (z)={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {(1+1/n)^{z}}{1+z/n}}} compute 1 Γ ( 1 − z ) Γ ( z ) = 1 ( − z ) Γ ( − z ) Γ ( z ) = z ∏ n = 1 ∞ ( 1 − z / n ) ( 1 + z / n ) ( 1 + 1 / n ) − z ( 1 + 1 / n ) z = z ∏ n = 1 ∞ ( 1 − z 2 n 2 ) = sin π z π , {\displaystyle {\frac {1}{\Gamma (1-z)\Gamma (z)}}={\frac {1}{(-z)\Gamma (-z)\Gamma (z)}}=z\prod _{n=1}^{\infty }{\frac {(1-z/n)(1+z/n)}{(1+1/n)^{-z}(1+1/n)^{z}}}=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)={\frac {\sin \pi z}{\pi }}\,,} where
SECTION 50
#17327810120614784-563: The double factorial ( 2 n − 1 ) ! ! = ( 2 n − 1 ) ( 2 n − 3 ) ⋯ ( 3 ) ( 1 ) {\displaystyle (2n-1)!!=(2n-1)(2n-3)\cdots (3)(1)} . See Particular values of the gamma function for calculated values. It might be tempting to generalize the result that Γ ( 1 2 ) = π {\textstyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}} by looking for
4876-656: The gamma function (represented by Γ, capital Greek letter gamma ) is the most common extension of the factorial function to complex numbers . Derived by Daniel Bernoulli , the gamma function Γ ( z ) {\displaystyle \Gamma (z)} is defined for all complex numbers z {\displaystyle z} except non-positive integers, and for every positive integer z = n {\displaystyle z=n} , Γ ( n ) = ( n − 1 ) ! . {\displaystyle \Gamma (n)=(n-1)!\,.} The gamma function can be defined via
4968-425: The half-plane of absolute convergence : where the product is over all prime numbers . 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. This is because of the relationship between a imprimitive character χ {\displaystyle \chi } and
5060-562: The multiplication theorem (see Eq. 5.5.6): ∏ k = 0 m − 1 Γ ( z + k m ) = ( 2 π ) m − 1 2 m 1 2 − m z Γ ( m z ) . {\displaystyle \prod _{k=0}^{m-1}\Gamma \left(z+{\frac {k}{m}}\right)=(2\pi )^{\frac {m-1}{2}}\;m^{{\frac {1}{2}}-mz}\;\Gamma (mz).} A simple but useful property, which can be seen from
5152-414: The principal character , usually denoted χ 0 {\displaystyle \chi _{0}} , (see Notation below) exists for all moduli: The German mathematician Peter Gustav Lejeune Dirichlet —for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions . ϕ ( n ) {\displaystyle \phi (n)}
5244-411: The residue theorem , ∫ C R e a z 1 + e z d z = − 2 π i e a π i . {\displaystyle \int _{C_{R}}{\frac {e^{az}}{1+e^{z}}}\,dz=-2\pi ie^{a\pi i}.} Let I R = ∫ − R R e
5336-443: The theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L ( s , χ ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L -function has a simple pole at s = 1. Otherwise, the L -function is entire . Since a Dirichlet character χ is completely multiplicative , its L -function can also be written as an Euler product in
5428-515: 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: Dirichlet character In analytic number theory and related branches of mathematics,
5520-460: The Legendre duplication formula follows: Γ ( z ) Γ ( z + 1 2 ) = 2 1 − 2 z π Γ ( 2 z ) . {\displaystyle \Gamma (z)\Gamma \left(z+{\frac {1}{2}}\right)=2^{1-2z}{\sqrt {\pi }}\;\Gamma (2z).} The duplication formula is a special case of
5612-539: 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 for β + iγ a non-real zero. The Dirichlet L -functions may be written as a linear combination of the Hurwitz zeta function at rational values. Fixing an integer k ≥ 1, the Dirichlet L -functions for characters modulo k are linear combinations, with constant coefficients, of
SECTION 60
#17327810120615704-5069: The Weierstrass definition and Euler definition Γ ( z ) = e − γ z z ∏ n = 1 ∞ ( 1 + z n ) − 1 e z / n = 1 z lim n → ∞ e z ( log n − 1 − 1 2 − 1 3 − ⋯ − 1 n ) e z ( 1 + 1 2 + 1 3 + ⋯ + 1 n ) ( 1 + z ) ( 1 + z 2 ) ⋯ ( 1 + z n ) = 1 z lim n → ∞ 1 ( 1 + z ) ( 1 + z 2 ) ⋯ ( 1 + z n ) e z log ( n ) = lim n → ∞ n ! n z z ( z + 1 ) ⋯ ( z + n ) , z ∈ C ∖ Z 0 − {\displaystyle {\begin{aligned}\Gamma (z)&={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n}\\&={\frac {1}{z}}\lim _{n\to \infty }e^{z\left(\log n-1-{\frac {1}{2}}-{\frac {1}{3}}-\cdots -{\frac {1}{n}}\right)}{\frac {e^{z\left(1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}\right)}}{\left(1+z\right)\left(1+{\frac {z}{2}}\right)\cdots \left(1+{\frac {z}{n}}\right)}}\\&={\frac {1}{z}}\lim _{n\to \infty }{\frac {1}{\left(1+z\right)\left(1+{\frac {z}{2}}\right)\cdots \left(1+{\frac {z}{n}}\right)}}e^{z\log \left(n\right)}\\&=\lim _{n\to \infty }{\frac {n!n^{z}}{z(z+1)\cdots (z+n)}},\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}\end{aligned}}} Let Γ n ( z ) = n ! n z z ( z + 1 ) ⋯ ( z + n ) {\displaystyle \Gamma _{n}(z)={\frac {n!n^{z}}{z(z+1)\cdots (z+n)}}} and G n ( z ) = ( n − 1 ) ! n z z ( z + 1 ) ⋯ ( z + n − 1 ) . {\displaystyle G_{n}(z)={\frac {(n-1)!n^{z}}{z(z+1)\cdots (z+n-1)}}.} Then Γ n ( z ) = n z + n G n ( z ) {\displaystyle \Gamma _{n}(z)={\frac {n}{z+n}}G_{n}(z)} and lim n → ∞ G n + 1 ( z ) = lim n → ∞ G n ( z ) = lim n → ∞ Γ n ( z ) = Γ ( z ) , {\displaystyle \lim _{n\to \infty }G_{n+1}(z)=\lim _{n\to \infty }G_{n}(z)=\lim _{n\to \infty }\Gamma _{n}(z)=\Gamma (z),} therefore Γ ( z ) = lim n → ∞ n ! ( n + 1 ) z z ( z + 1 ) ⋯ ( z + n ) , z ∈ C ∖ Z 0 − . {\displaystyle \Gamma (z)=\lim _{n\to \infty }{\frac {n!(n+1)^{z}}{z(z+1)\cdots (z+n)}},\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}.} Then n ! ( n + 1 ) z z ( z + 1 ) ⋯ ( z + n ) = ( 2 / 1 ) z ( 3 / 2 ) z ( 4 / 3 ) z ⋯ ( ( n + 1 ) / n ) z z ( 1 + z ) ( 1 + z / 2 ) ( 1 + z / 3 ) ⋯ ( 1 + z / n ) = 1 z ∏ k = 1 n ( 1 + 1 / k ) z 1 + z / k , z ∈ C ∖ Z 0 − {\displaystyle {\frac {n!(n+1)^{z}}{z(z+1)\cdots (z+n)}}={\frac {(2/1)^{z}(3/2)^{z}(4/3)^{z}\cdots ((n+1)/n)^{z}}{z(1+z)(1+z/2)(1+z/3)\cdots (1+z/n)}}={\frac {1}{z}}\prod _{k=1}^{n}{\frac {(1+1/k)^{z}}{1+z/k}},\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}} and taking n → ∞ {\displaystyle n\to \infty } gives
5796-456: The ambiguity is the Bohr–Mollerup theorem , which shows that f ( x ) = Γ ( x ) {\displaystyle f(x)=\Gamma (x)} is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex , meaning that y = log f ( x ) {\displaystyle y=\log f(x)}
5888-583: The base case is that Γ ( 1 ) = 1 = 0 ! {\displaystyle \Gamma (1)=1=0!} , and the induction step is that Γ ( n + 1 ) = n Γ ( n ) = n ( n − 1 ) ! = n ! . {\displaystyle \Gamma (n+1)=n\Gamma (n)=n(n-1)!=n!.} The identity Γ ( z ) = Γ ( z + 1 ) z {\textstyle \Gamma (z)={\frac {\Gamma (z+1)}{z}}} can be used (or, yielding
5980-431: The characters mod 3 are 2 is a primitive root mod 5. ( ϕ ( 5 ) = 4 {\displaystyle \phi (5)=4} ) so the values of ν 5 {\displaystyle \nu _{5}} are The nonzero values of the characters mod 5 are 3 is a primitive root mod 7. ( ϕ ( 7 ) = 6 {\displaystyle \phi (7)=6} ) so
6072-1061: The desired result. Besides the fundamental property discussed above: Γ ( z + 1 ) = z Γ ( z ) {\displaystyle \Gamma (z+1)=z\ \Gamma (z)} other important functional equations for the gamma function are Euler's reflection formula Γ ( 1 − z ) Γ ( z ) = π sin π z , z ∉ Z {\displaystyle \Gamma (1-z)\Gamma (z)={\frac {\pi }{\sin \pi z}},\qquad z\not \in \mathbb {Z} } which implies Γ ( z − n ) = ( − 1 ) n − 1 Γ ( − z ) Γ ( 1 + z ) Γ ( n + 1 − z ) , n ∈ Z {\displaystyle \Gamma (z-n)=(-1)^{n-1}\;{\frac {\Gamma (-z)\Gamma (1+z)}{\Gamma (n+1-z)}},\qquad n\in \mathbb {Z} } and
6164-451: The factorial, x ! = 1 × 2 × ⋯ × x is only valid when x is a positive integer, and no elementary function has this property, but a good solution is the gamma function f ( x ) = Γ ( x + 1 ) {\displaystyle f(x)=\Gamma (x+1)} . The gamma function is not only smooth but analytic (except at the non-positive integers), and it can be defined in several explicit ways. However, it
6256-540: The fields of probability , statistics , analytic number theory , and combinatorics . The gamma function can be seen as a solution to the interpolation problem of finding a smooth curve y = f ( x ) {\displaystyle y=f(x)} that connects the points of the factorial sequence: ( x , y ) = ( n , n ! ) {\displaystyle (x,y)=(n,n!)} for all positive integer values of n {\displaystyle n} . The simple formula for
6348-651: The functional equation is: In this equation, Γ denotes the gamma function ; where τ ( χ ) is a Gauss sum : It is a property of Gauss sums that | τ ( χ ) | = q , so | W ( χ ) | = 1. Another way to state the functional equation is in terms of The functional equation can be expressed as: The functional equation implies that L ( s , χ ) {\displaystyle L(s,\chi )} (and Λ ( s , χ ) {\displaystyle \Lambda (s,\chi )} ) are entire functions of s . (Again, this assumes that χ
6440-416: 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 generalized Riemann hypothesis is the conjecture that all the non-trivial zeros lie on the critical line Re( s ) = 1/2. Up to the possible existence of a Siegel zero , zero-free regions including and beyond the line Re( s ) = 1 similar to that of
6532-497: The index t {\displaystyle t} is described in the section the group of characters below. In this labeling, χ m , _ ( a ) {\displaystyle \chi _{m,\_}(a)} denotes an unspecified character and χ m , 1 ( a ) {\displaystyle \chi _{m,1}(a)} denotes the principal character mod m {\displaystyle m} . The word " character "
6624-464: The last equality is a known result . A similar derivation begins with Weierstrass's definition. Proof 2 First prove that I = ∫ − ∞ ∞ e a x 1 + e x d x = ∫ 0 ∞ v a − 1 1 + v d v = π sin π
6716-437: The limit definition, is: Γ ( z ) ¯ = Γ ( z ¯ ) ⇒ Γ ( z ) Γ ( z ¯ ) ∈ R . {\displaystyle {\overline {\Gamma (z)}}=\Gamma ({\overline {z}})\;\Rightarrow \;\Gamma (z)\Gamma ({\overline {z}})\in \mathbb {R} .} In particular, with z =
6808-491: The modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of Conrey labeling (introduced by Brian Conrey and used by the LMFDB ). In this labeling characters for modulus m {\displaystyle m} are denoted χ m , t ( a ) {\displaystyle \chi _{m,t}(a)} where
6900-429: The negative exponential function : Γ ( z ) = M { e − x } ( z ) . {\displaystyle \Gamma (z)={\mathcal {M}}\{e^{-x}\}(z)\,.} Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It appears as a factor in various probability-distribution functions and other formulas in
6992-448: The nonzero values of χ ( a ) {\displaystyle \chi (a)} are ϕ ( m ) {\displaystyle \phi (m)} -th roots of unity : for some integer r {\displaystyle r} which depends on χ , ζ , {\displaystyle \chi ,\zeta ,} and a {\displaystyle a} . This implies there are only
7084-419: The orthogonality relations: The elements of the finite abelian group ( Z / m Z ) × {\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }} are the residue classes [ a ] = { x : x ≡ a ( mod m ) } {\displaystyle [a]=\{x:x\equiv a{\pmod {m}}\}} where (
7176-473: The primitive character χ ⋆ {\displaystyle \chi ^{\star }} which induces it: (Here, q is the modulus of χ .) An application of the Euler product gives a simple relationship between the corresponding L -functions: (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
7268-2546: The real part is an integer or a half-integer, this can be finitely expressed in closed form : | Γ ( b i ) | 2 = π b sinh π b | Γ ( 1 2 + b i ) | 2 = π cosh π b | Γ ( 1 + b i ) | 2 = π b sinh π b | Γ ( 1 + n + b i ) | 2 = π b sinh π b ∏ k = 1 n ( k 2 + b 2 ) , n ∈ N | Γ ( − n + b i ) | 2 = π b sinh π b ∏ k = 1 n ( k 2 + b 2 ) − 1 , n ∈ N | Γ ( 1 2 ± n + b i ) | 2 = π cosh π b ∏ k = 1 n ( ( k − 1 2 ) 2 + b 2 ) ± 1 , n ∈ N {\displaystyle {\begin{aligned}|\Gamma (bi)|^{2}&={\frac {\pi }{b\sinh \pi b}}\\[1ex]\left|\Gamma \left({\tfrac {1}{2}}+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\\[1ex]\left|\Gamma \left(1+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\\[1ex]\left|\Gamma \left(1+n+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right),\quad n\in \mathbb {N} \\[1ex]\left|\Gamma \left(-n+bi\right)\right|^{2}&={\frac {\pi }{b\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right)^{-1},\quad n\in \mathbb {N} \\[1ex]\left|\Gamma \left({\tfrac {1}{2}}\pm n+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\prod _{k=1}^{n}\left(\left(k-{\tfrac {1}{2}}\right)^{2}+b^{2}\right)^{\pm 1},\quad n\in \mathbb {N} \\[-1ex]&\end{aligned}}} First, consider
7360-507: The real part is integer or half-integer quickly follow by induction using the recurrence relation in the positive and negative directions. Perhaps the best-known value of the gamma function at a non-integer argument is Γ ( 1 2 ) = π , {\displaystyle \Gamma \left({\tfrac {1}{2}}\right)={\sqrt {\pi }},} which can be found by setting z = 1 2 {\textstyle z={\frac {1}{2}}} in
7452-414: The reciprocal 1 Γ ( z ) = z ∏ n = 1 ∞ [ ( 1 + z n ) / ( 1 + 1 n ) z ] {\displaystyle {\frac {1}{\Gamma (z)}}=z\prod _{n=1}^{\infty }\left[\left(1+{\frac {z}{n}}\right)/{\left(1+{\frac {1}{n}}\right)^{z}}\right]}
7544-1034: The reflection formula applied to z = 1 2 + b i {\displaystyle z={\tfrac {1}{2}}+bi} . Γ ( 1 2 + b i ) Γ ( 1 − ( 1 2 + b i ) ) = Γ ( 1 2 + b i ) Γ ( 1 2 − b i ) = π sin π ( 1 2 + b i ) = π cos π b i = π cosh π b {\displaystyle \Gamma ({\tfrac {1}{2}}+bi)\Gamma \left(1-({\tfrac {1}{2}}+bi)\right)=\Gamma ({\tfrac {1}{2}}+bi)\Gamma ({\tfrac {1}{2}}-bi)={\frac {\pi }{\sin \pi ({\tfrac {1}{2}}+bi)}}={\frac {\pi }{\cos \pi bi}}={\frac {\pi }{\cosh \pi b}}} Formulas for other values of z {\displaystyle z} for which
7636-1114: The reflection formula applied to z = b i {\displaystyle z=bi} . Γ ( b i ) Γ ( 1 − b i ) = π sin π b i {\displaystyle \Gamma (bi)\Gamma (1-bi)={\frac {\pi }{\sin \pi bi}}} Applying the recurrence relation to the second term: − b i ⋅ Γ ( b i ) Γ ( − b i ) = π sin π b i {\displaystyle -bi\cdot \Gamma (bi)\Gamma (-bi)={\frac {\pi }{\sin \pi bi}}} which with simple rearrangement gives Γ ( b i ) Γ ( − b i ) = π − b i sin π b i = π b sinh π b {\displaystyle \Gamma (bi)\Gamma (-bi)={\frac {\pi }{-bi\sin \pi bi}}={\frac {\pi }{b\sinh \pi b}}} Second, consider
7728-1408: The reflection formula for all z ∈ ( 0 , 1 ) {\displaystyle z\in (0,1)} proves it for all z ∈ C ∖ Z {\displaystyle z\in \mathbb {C} \setminus \mathbb {Z} } by analytic continuation. The beta function can be represented as B ( z 1 , z 2 ) = Γ ( z 1 ) Γ ( z 2 ) Γ ( z 1 + z 2 ) = ∫ 0 1 t z 1 − 1 ( 1 − t ) z 2 − 1 d t . {\displaystyle \mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt.} Setting z 1 = z 2 = z {\displaystyle z_{1}=z_{2}=z} yields Γ 2 ( z ) Γ ( 2 z ) = ∫ 0 1 t z − 1 ( 1 − t ) z − 1 d t . {\displaystyle {\frac {\Gamma ^{2}(z)}{\Gamma (2z)}}=\int _{0}^{1}t^{z-1}(1-t)^{z-1}\,dt.} After
7820-405: The reflection or duplication formulas, by using the relation to the beta function given below with z 1 = z 2 = 1 2 {\textstyle z_{1}=z_{2}={\frac {1}{2}}} , or simply by making the substitution u = z {\displaystyle u={\sqrt {z}}} in the integral definition of the gamma function, resulting in
7912-490: The relationship Γ ( x + 1 ) = x Γ ( x ) {\displaystyle \Gamma (x+1)=x\Gamma (x)} backwards n + 1 {\displaystyle n+1} times to get an approximation for Γ ( z ) {\displaystyle \Gamma (z)} ; and furthermore that this approximation becomes exact as n {\displaystyle n} increases to infinity. The infinite product for
8004-418: The same result, analytic continuation can be used) to uniquely extend the integral formulation for Γ ( z ) {\displaystyle \Gamma (z)} to a meromorphic function defined for all complex numbers z , except integers less than or equal to zero. It is this extended version that is commonly referred to as the gamma function. There are many equivalent definitions. For
8096-698: The substitution t = 1 + u 2 {\displaystyle t={\frac {1+u}{2}}} : Γ 2 ( z ) Γ ( 2 z ) = 1 2 2 z − 1 ∫ − 1 1 ( 1 − u 2 ) z − 1 d u . {\displaystyle {\frac {\Gamma ^{2}(z)}{\Gamma (2z)}}={\frac {1}{2^{2z-1}}}\int _{-1}^{1}\left(1-u^{2}\right)^{z-1}\,du.} The function ( 1 − u 2 ) z − 1 {\displaystyle (1-u^{2})^{z-1}}
8188-510: The trivial zeros. 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
8280-399: The units ≡ 3 ( mod 4 ) . {\displaystyle \equiv 3{\pmod {4}}.} For example Let q = 2 k , k ≥ 3 {\displaystyle q=2^{k},\;\;k\geq 3} ; then ( Z / q Z ) × {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}
8372-454: The values of ν 7 {\displaystyle \nu _{7}} are The nonzero values of the characters mod 7 are ( ω = ζ 6 , ω 3 = − 1 {\displaystyle \omega =\zeta _{6},\;\;\omega ^{3}=-1} ) 2 is a primitive root mod 9. ( ϕ ( 9 ) = 6 {\displaystyle \phi (9)=6} ) so
8464-456: The values of ν 9 {\displaystyle \nu _{9}} are The nonzero values of the characters mod 9 are ( ω = ζ 6 , ω 3 = − 1 {\displaystyle \omega =\zeta _{6},\;\;\omega ^{3}=-1} ) ( Z / 2 Z ) × {\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{\times }}
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