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 ${\displaystyle s}$ a complex variable with real part greater than ${\displaystyle 1}$. 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; it is then called a Dirichlet L-function.

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in 1837 ^[1] to prove his theorem on primes in arithmetic progressions. In his proof, Dirichlet showed that ${\displaystyle L(s,\chi )}$ is non-zero at ${\displaystyle s=1}$. Moreover, if ${\displaystyle \chi }$ is principal, then the corresponding Dirichlet L-function has a simple pole at ${\displaystyle s=1}$. Otherwise, the L-function is entire.

Euler product

Since a Dirichlet character ${\displaystyle \chi }$ 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. ^[2]

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. ^[3] This is because of the relationship between a imprimitive character ${\displaystyle \chi }$ and the primitive character ${\displaystyle \chi ^{\star }}$ which induces it: ^[4]

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

(Here, ${\displaystyle q}$ is the modulus of ${\displaystyle \chi }$.) An application of the Euler product gives a simple relationship between the corresponding L-functions: ^{[5] [6]}

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

By analytic continuation, this formula holds for all complex ${\displaystyle s}$, even though the Euler product is only valid when ${\displaystyle \operatorname {Re} (s)>1}$. The formula shows that the L-function of ${\displaystyle \chi }$ is equal to the L-function of the primitive character which induces ${\displaystyle \chi }$, multiplied by only a finite number of factors. ^[7]

As a special case, the L-function of the principal character ${\displaystyle \chi _{0}}$ modulo ${\displaystyle q}$ can be expressed in terms of the Riemann zeta function: ^{[8] [9]}

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

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 values of ${\displaystyle L(s,\chi )}$ to the values of ${\displaystyle L(1-s,{\overline {\chi }})}$.

Let ${\displaystyle \chi }$ be a primitive character modulo ${\displaystyle q}$, where ${\displaystyle q>1}$. One way to express the functional equation is as ^[10]

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

where ${\displaystyle \Gamma }$ is the gamma function, ${\displaystyle \chi (-1)=(-1)^{\delta }}$, and

${\displaystyle W(\chi )={\frac {\tau (\chi )}{i^{\delta }{\sqrt {q}}}},}$

where ${\displaystyle \tau (\chi )}$ is the Gauss sum

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

It is a property of Gauss sums that ${\displaystyle |\tau (\chi )|={\sqrt {q}}}$, so ${\displaystyle |W(\chi )|=1}$. ^{[11] [12]} Another functional equation is

${\displaystyle \Lambda (s,\chi )=q^{s/2}\pi ^{-(s+\delta )/2}\operatorname {\Gamma } \left({\frac {s+\delta }{2}}\right)L(s,\chi ),}$

which can be expressed as ^{[10] [12]}

${\displaystyle \Lambda (s,\chi )=W(\chi )\Lambda (1-s,{\overline {\chi }}).}$

This implies that ${\displaystyle L(s,\chi )}$ and ${\displaystyle \Lambda (s,\chi )}$ are entire functions of ${\displaystyle s}$. Again, this assumes that ${\displaystyle \chi }$ is primitive character modulo ${\displaystyle q}$ with ${\displaystyle q>1}$. If ${\displaystyle q=1}$, then ${\displaystyle L(s,\chi )=\zeta (s)}$ has a pole at ${\displaystyle s=1}$. ^{[10] [12]}

For generalizations, see the article on functional equations of L-functions.

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 ${\displaystyle \chi }$ be a primitive character modulo ${\displaystyle q}$, with ${\displaystyle q>1}$.

There are no zeros of ${\displaystyle L(s,\chi )}$ with ${\displaystyle \operatorname {Re} (s)>1}$. For ${\displaystyle \operatorname {Re} (s)<0}$, there are zeros at certain negative integers ${\displaystyle s}$:

• If ${\displaystyle \chi (-1)=1}$, the only zeros of ${\displaystyle L(s,\chi )}$ with ${\displaystyle \operatorname {Re} (s)<0}$ are simple zeros at ${\displaystyle -2,-4,-6,\dots }$ There is also a zero at ${\displaystyle s=0}$ when ${\displaystyle \chi }$ is non-principal. These correspond to the poles of ${\displaystyle \textstyle \Gamma ({\frac {s}{2}})}$. ^[13]
• If ${\displaystyle \chi (-1)=-1}$, then the only zeros of ${\displaystyle L(s,\chi )}$ with ${\displaystyle \operatorname {Re} (s)<0}$ are simple zeros at ${\displaystyle -1,-3,-5,\dots }$ These correspond to the poles of ${\displaystyle \textstyle \Gamma ({\frac {s+1}{2}})}$. ^[13]

These are called the trivial zeros. ^[10]

The remaining zeros lie in the critical strip ${\displaystyle 0\leq \operatorname {Re} (s)\leq 1}$, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line ${\displaystyle \operatorname {Re} (s)=1/2}$. That is, if ${\displaystyle L(\rho ,\chi )=0}$, then ${\displaystyle L(1-{\overline {\rho }},\chi )=0}$ too because of the functional equation. If ${\displaystyle \chi }$ is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if ${\displaystyle \chi }$ is a complex character. The generalized Riemann hypothesis is the conjecture that all the non-trivial zeros lie on the critical line ${\displaystyle \operatorname {Re} (s)=1/2}$. ^[10]

Up to the possible existence of a Siegel zero, zero-free regions including and beyond the line ${\displaystyle \operatorname {Re} (s)=1}$ similar to that of the Riemann zeta function are known to exist for all Dirichlet L-functions: for example, for ${\displaystyle \chi }$ a non-real character of modulus ${\displaystyle q}$, we have

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

for ${\displaystyle \beta +i\gamma }$ a non-real zero. ^[14]

Relation to the Hurwitz zeta function

Dirichlet L-functions may be written as linear combinations of the Hurwitz zeta function at rational values. Fixing an integer ${\displaystyle k\geq 1}$, Dirichlet L-functions for characters modulo ${\displaystyle k}$ are linear combinations with constant coefficients of the ${\displaystyle \zeta (s,a)}$ where ${\displaystyle a=r/k}$ and ${\displaystyle r=1,2,\dots ,k}$. This means that the Hurwitz zeta function for rational ${\displaystyle a}$ has analytic properties that are closely related to the Dirichlet L-functions. Specifically, if ${\displaystyle \chi }$ is a character modulo ${\displaystyle k}$, we can write its Dirichlet L-function as ^[15]

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

See also

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

Notes

1. Dirichlet, Peter Gustav Lejeune (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.
2. Apostol 1976 , Theorem 11.7
3. Davenport 2000 , chapter 5
4. Davenport 2000 , chapter 5, equation (2)
5. Davenport 2000 , chapter 5, equation (3)
6. Montgomery & Vaughan 2006 , p. 282
7. Apostol 1976 , p. 262
8. Ireland & Rosen 1990 , chapter 16, section 4
9. Montgomery & Vaughan 2006 , p. 121
10. Montgomery & Vaughan 2006 , p. 333
11. Montgomery & Vaughan 2006 , p. 332
12. Iwaniec & Kowalski 2004 , p. 84
13. Davenport 2000 , chapter 9
14. 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.
15. 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]