Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form

${\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.}$

where ${\displaystyle \chi }$ 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 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.

Euler product

Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product in the half-plane of absolute convergence:

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

where the product is over all prime numbers. ^[1]

Primitive characters

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. ^[2] This is because of the relationship between a imprimitive character ${\displaystyle \chi }$ and the primitive character ${\displaystyle \chi ^{\star }}$ which induces it: ^[3]

${\displaystyle \chi (n)={\begin{cases}\chi ^{\star }(n),&\mathrm {if} \gcd(n,q)=1\\0,&\mathrm {if} \gcd(n,q)\neq 1\end{cases}}}$

(Here, q is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-functions: ^{[4] [5]}

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

(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. ^[6]

As a special case, the L-function of the principal character ${\displaystyle \chi _{0}}$ modulo q can be expressed in terms of the Riemann zeta function: ^{[7] [8]}

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

Functional equation

Dirichlet L-functions satisfy a functional equation, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of ${\displaystyle L(s,\chi )}$ to the value of ${\displaystyle L(1-s,{\overline {\chi }})}$. Let χ be a primitive character modulo q, where q > 1. One way to express the functional equation is: ^[9]

${\displaystyle L(s,\chi )=\varepsilon (\chi )2^{s}\pi ^{s-1}q^{1/2-s}\sin \left({\frac {\pi }{2}}(s+a)\right)\Gamma (1-s)L(1-s,{\overline {\chi }}).}$

In this equation, Γ denotes the gamma function; a is 0 if χ(−1) = 1, or 1 if χ(−1) = −1; and

${\displaystyle \varepsilon (\chi )={\frac {\tau (\chi )}{i^{a}{\sqrt {q}}}}}$

where τ ( χ) is a Gauss sum:

${\displaystyle \tau (\chi )=\sum _{n=1}^{q}\chi (n)\exp(2\pi in/q).}$

It is a property of Gauss sums that |τ ( χ) | = q^1/2, so |ɛ ( χ) | = 1. ^{[10] [11]}

Another way to state the functional equation is in terms of

${\displaystyle \xi (s,\chi )=\left({\frac {q}{\pi }}\right)^{(s+a)/2}\operatorname {\Gamma } \left({\frac {s+a}{2}}\right)L(s,\chi ).}$

The functional equation can be expressed as: ^{[9] [11]}

${\displaystyle \xi (s,\chi )=\varepsilon (\chi )\xi (1-s,{\overline {\chi }}).}$

The functional equation implies that ${\displaystyle L(s,\chi )}$ (and ${\displaystyle \xi (s,\chi )}$) are entire functions of s. (Again, this assumes that χ is primitive character modulo q with q > 1. If q = 1, then ${\displaystyle L(s,\chi )=\zeta (s)}$ has a pole at s = 1.) ^{[9] [11]}

For generalizations, see: Functional equation (L-function).

Zeros

The Dirichlet L-function L(s, χ) = 1 − 3 + 5 − 7 + ⋅⋅⋅ (sometimes given the special name Dirichlet beta function), with trivial zeros at the negative odd integers

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:

• 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 ${\displaystyle \textstyle \Gamma ({\frac {s}{2}})}$. ^[12]
• 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 ${\displaystyle \textstyle \Gamma ({\frac {s+1}{2}})}$. ^[12]

These are called the trivial zeros. ^[9]

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 ${\displaystyle L(\rho ,\chi )=0}$ then ${\displaystyle L(1-{\overline {\rho }},\chi )=0}$ 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 generalized Riemann hypothesis is the conjecture that all the non-trivial zeros lie on the critical line Re(s) = 1/2. ^[9]

Up to the possible existence of a Siegel zero, 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

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

for β + iγ a non-real zero. ^[13]

Relation to the Hurwitz zeta function

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 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: ^[14]

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

See also

• Generalized Riemann hypothesis
• L-function
• Modularity theorem
• Artin conjecture
• Special values of L-functions

Notes

1. Apostol 1976 , Theorem 11.7
2. Davenport 2000 , chapter 5
3. Davenport 2000 , chapter 5, equation (2)
4. Davenport 2000 , chapter 5, equation (3)
5. Montgomery & Vaughan 2006 , p. 282
6. Apostol 1976 , p. 262
7. Ireland & Rosen 1990 , chapter 16, section 4
8. Montgomery & Vaughan 2006 , p. 121
9. Montgomery & Vaughan 2006 , p. 333
10. Montgomery & Vaughan 2006 , p. 332
11. Iwaniec & Kowalski 2004 , p. 84
12. Davenport 2000 , chapter 9
13. 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.
14. Apostol 1976 , p. 249

References

• 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
• Apostol, T. M. (2010), "Dirichlet L-function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN   978-0-521-19225-5, MR   2723248 .
• Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer. ISBN   0-387-95097-4.
• Dirichlet, P. G. L. (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.
• Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory (2nd ed.). Springer-Verlag.
• 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.
• Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic Number Theory. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society.
• "Dirichlet-L-function", Encyclopedia of Mathematics , EMS Press, 2001 [1994]