A history of L-functions (reviewed)

The prototype of L-functions is the Riemann zeta-function defined by \[ \zeta(s)=\sum_{n=1}^\infty \frac {1}{n^s} \] for $\Re s>1$. Euler discovered that $\zeta(2)=\frac{\pi^2}{6}$, and he also found the fundamental identity \[ \zeta(s)=\prod_p \left(1-\frac{1}{p^s}\right)^{-1}, \] which he used to prove that the series $\sum_p \frac{1}{p}$ diverges. Euler considered other divergent series and noticed a connection between values of $\zeta$ at $s$ and $1-s$, linked by powers of $\pi$ and Bernoulli numbers, such as the interpretations $$1+2+3+\dots =-\frac{1}{12}$$ and $$1+4+9+16+\dots =0.$$ Euler found hints of many of the remarkable features of L-functions that make them such worthy objects of study: the Euler product, special values, and the functional equation. It was left to Riemann to discover perhaps the most remarkable property of all, one that hasn't been proven yet: the Riemann Hypothesis that each non-real zero of $\zeta(s)$ has real part equal to $\frac 12 $.

Riemann's memoir

In 1859, in an 8 page paper “Über die Anzahl der Primzahlen unter einer gegebenen Grösse” [Transcript] read to the Berlin Academy of Sciences by Gauss' former student Encke, Riemann first considered $\zeta(s)$ as a function of a complex variable $s$. He proved that $\zeta(s)$ is a meromorphic function in the complex $s$-plane whose only singularity is a simple pole at $s=1$ with residue 1. He went on to prove that $$\xi(s)=\frac{1}{2} s(s-1)\pi^{-s/2}\Gamma(s/2) \zeta(s)$$ is entire of order 1 and satisfies the functional equation $$\xi(s)=\xi(1-s).$$ He proved that $\xi(s)$ has infinitely many zeros, all of which are in the critical strip $0\le \sigma\le 1$, where $\sigma$ denotes the real part of $s$. Riemann calculated the first few of these zeros (this was discovered later when Siegel studied Riemann's notes left to the Göttingen library) and found them to lie on the critical line $\sigma=1/2$. He conjectured that all of the zeros are on this line. And the analytic theory of L-functions was born!

Dirichlet L-functions

The first use of the letter L to denote these functions was by Dirichlet in 1837 (see Werke I [MR:249268] pages 313-342) who used L-functions to prove that there are infinitely many primes in any (primitive) arithmetic progression. He considered series of the form $$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ where $\chi$ is a (Dirichlet) character, which is the extension to the integers of a character of the group $(\mathbb Z/q\mathbb Z)^*$. For example the arithmetic function $\chi$ defined by

\[\chi(n)=\begin{cases} 1 & \textrm{ if } n\equiv 1 \bmod 3\\ -1 & \textrm{ if } n\equiv 2\bmod 3\\ 0 & \textrm{ if }n\equiv 0\bmod 3\end{cases}\]

is a Dirichlet character modulo 3. The above L-function has an Euler product $$L(s,\chi)=\prod_{p }\left(1-\frac{\chi(p)}{p^s}\right)^{-1}$$ and satisfies a functional equation that $$\left(\frac{3}{\pi}\right)^{\frac{s}{2}}\Gamma\left(\frac{s+1}{2}\right) L(s,\chi)$$ is invariant under $s\to 1-s$. Also, note the special value $$L(1,\chi)= \frac{\pi}{3\sqrt{3}}; $$ Dirichlet needed to know that his L-functions did not vanish at 1 and he used special values to prove this fact. Dirichlet's original proof was for prime moduli; for composite moduli he required his class number formula which was proven in 1839 - 1840 (see his Werke I, pp. 411-496).

Dedekind zeta functions

In 1877 Dedekind began generalizing some of Dirichlet's work to number fields. His first paper on the topic is Über die Anzahl der Ideal-Klassen in den verschiedenen Ordnungen eines endlichen Körpers [MR:237282]. It appears that Hecke named the Dedekind zeta-function after him.

The automorphy properties of Hilbert modular forms for real quadratic fields were considered in the 1901 Göttingen University Habilitationssschrift of Otto Blumenthal. Therein he refers to unpublished work of 1893-1894 of his advisor David Hilbert. Hilbert modular forms were significantly developed by Hecke in his 1910 dissertation.

The L-function of $\Delta$

In 1916 Ramanujan [MR:2280861] made the startling observation that the coefficients of the $\Delta$ function are multiplicative! Ramanujan defined $\tau(n)$ by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau(n)q^n$$ and conjectured that $\tau(mn)=\tau(m)\tau(n)$ whenever $m$ and $n$ are relatively prime. Moreover that $$\sum_{r=0}^\infty \tau(p^r)X^r = (1-\tau(p)X +p^{11}X^2)^{-1}$$ and that $|\tau(p)|< 2p^{11/2}$. The first two of these astounding conjectures were verified by Mordell in 1917 (see “On Mr. Ramanujan's Empirical Expansions of Modular Functions.” Proc. Cambridge Phil. Soc.19, (1917)) and the last by Deligne in 1974 in work for which he won a Fields Medal. It was already known that $$\Delta(z)=\sum_{n=1}^\infty \tau(n)e(nz)$$ is a modular form of weight 12. Thus $L(s)=\sum_{n=1}^\infty \tau(n)n^{-s}$ has a functional equation and (by Mordell) an Euler product. Ramanujan's discovery was the ushering in of a new age of arithmetic L-functions.

Hecke L-functions

The L-functions associated with finite characters of number fields (often called Dirichlet charaters or finite Hecke characters) were considered by Hecke in a series of papers beginning in 1917. In the first paper of this series, Über die Zetafunktion beliebiger algebraischer Zahlkörper [G&ouml;tt. Nachr. 1917 77-89 (1917)], he refers to $\zeta_k(s)$ as the "Dirichlet-Dedekindsche Zetafunktion". Here he gives the functional equation for the Dedekind zeta-function of any number field. In the second paper Über eine neue anwendung der Zetafunktionnen auf die arithmetik der Zahlkörper also in 1917, he refers to it as "der Dedekindschen Funktion $\zeta_k(s)$." This paper ends with the remark that for abelian extensions $\zeta_k(s)$ can be factored as a product of Dirichlet L-functions. In the third paper in the series, he refers again to "Dirichlet-Dedekindsche Funktion $\zeta_k(s)$". Here he proves the functional equation for L-functions of finite Hecke characters.

The L-functions associated with Hecke's Gröβencharaktere, i.e. characters of infinite order, and their functional equations appear in a two-paper series beginning in 1918 [MR:1544392]. It is only in the second of these papers (1920) that Hecke uses the term "Gröβencharaktere".

Hecke [MR:1513122], building on Mordell, introduced operators acting on vector spaces of modular forms. The forms which were simultaneous eigenvalues of these operators have multiplicative Fourier coefficients so that their associated Dirichlet series have Euler products and functional equations. Hecke mainly worked with level 1 modular forms. The more subtle theory of Hecke operators for spaces of higher level and with character was developed by Atkin and Lehner, and Li [MR:268123].

Artin L-functions

Artin L-functions were introduced in 1924 [MR:3069421]. Artin's conjecture that these L-functions are entire remains unsolved, though many instances are known.

Siegel Modular forms

In 1939 Siegel [MR:0001251] introduced his theory of higher degree modular forms and their L-functions.

Rankin-Selberg convolution

In 1939 R. A. Rankin [MR:0000411] studied Ramanujan's conjecture about the size of $\tau(n)$ and was led to consider the analytic properties of $g(s)=\sum_{n=1}^\infty \tau(n)^2 n^{-s}$. He proved that $\zeta(2s-22) g(s)$ is analytic everywhere except for a simple pole at $s=12$ and satisfies a functional equation. This was the beginning of the Rankin - Selberg convolution which we now realize was a hugely important event in the theory of L-functions. Selberg [MR:0002626] did the same calculation around the same time.

L-functions of nonholomorphic modular forms

In 1949 Hans Maass made his profound discovery that there are L-functions associated with non-holomorphic automorphic forms. In his Math Review of Maass' article [MR:0031519] J. Lehner writes

In Hecke's theory of Dirichlet series with Euler products we associate, roughly speaking, a Dirichlet series with an automorphic function; the invariance of the latter under linear substitutions is used, together with the Mellin transform, to derive a functional equation for the Dirichlet series. This suffices for the discussion of the $\zeta$-function of an imaginary quadratic field, for example, but not of a real quadratic field. In order to handle the latter case, the author defines a class of functions ("automorphic wave functions") to take the place of the analytic automorphic functions of Hecke's theory.

Maass' work prompted André Weil to remark “Il a fallu Maass pour nous sortir du ghetto des fonctions holomorphes.”

Tate's thesis

In 1950 Tate's PhD thesis “Fourier analysis in number fields and Hecke's zeta-functions” reprinted in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), 1967, pp. 305-347 Thompson, Washington, D.C., introduced a way to do harmonic analysis on adelic spaces and led to great simplifications in the calculations of Euler factors and functional equations of L-functions over number fields, in particular, and generally for any L-function with conductor larger than 1. This work paved the way for subsequent work associating L-functions to automorphic forms.

Hasse-Weil L-functions

In 1955 Hasse [MR:76807] introduced the zeta-function associated with a curve, today called the Hasse-Weil zeta function. For a Fermat curve $x^m+y^m=1$ he obtains an expression for his zeta-function in terms of L-functions with a Hecke character.

Langlands Program

Langlands theory of automorphic forms ushered in an age of a new understanding of the profundity of L-functions. His small book Euler Products of 1967 contains the beginnings of a general theory.

In 1969 Ogg [MR:0246819] proved the holomorphy and functional equation for the Rankin-Selberg convolution of two inequivalent cusp forms of the same level. This is a degree 4 L-function. In 1972 Jacquet [MR:0562503] considered the Rankin-Selberg convolution of two GL(2) cusp forms and obtained the analytic continuation and functional equation of the associated L-function. In 1979 Winnie Li [MR:0550843] completed the Rankin-Selberg story for two arbitrary GL(2) cusp forms. Her techniques really require the use of Tate's thesis and the theory of automorphic forms as it is virtually impossible to figure out the Euler factors at bad primes without these.

In 1971 Andrianov [MR:0340178] explicitly constructed the L-function for a genus 2 Siegel modular form and gave its analytic continuation and functional equation.

In 1972 Godement and Jacquet [MR:0342495] defined the L-function of a general automorphic form on a reductive group and obtained the analytic continuation and functional equation.

It wasn't until the mid 1970's that Shimura [MR:0382176] and, shortly later but independently, Zagier [MR:0485703] proved that the L-function associated with the symmetric square of a cusp form is entire. See also the footnote in Selberg's paper [MR:1220477] that suggests that Selberg had discovered this many years earlier.

In 1977 Asai [MR:0429751] obtained the holomorphy and functional equation of certain degree 4 L-functions associated with Hilbert modular forms over quadratic fields. These are known as Asai L-functions.

Beginning in 1978 Shahidi [MR:0498494] and completed in 1999 with Kim [MR:1726704] proved that the symmetric cube of a GL(2) L-function is entire and has functional equation.

In 1980 Langlands proved Artin's conjecture for 2-dimensional representations of tetrahedral type (and also for certain octahedral representations) [MR:0574808]. In 1981 Tunnell proved Artin's conjecture for octahedral representations [MR:0621884].

In work beginning in 1981 with [MR:0610479], Shahidi proved the holomorphy and functional equation for many automorphic L-functions.

In 1983 Jacquet, Piatetskii-Shapiro, and Shalika [MR:0701565]obtained the meromorphicity and functional equation for the L-function that is a general Rankin-Selberg convolution of automorphic L-functions.

In 1995 Andrew Wiles [MR:1333035] proved the holomorphy and functional equation of the Hasse-Weil zeta-function of most elliptic curves. In 2001 this work was extended by Breuil, Conrad, Diamond, and Taylor to include all elliptic curves [MR:1839918].