A study into the Selberg Classes.

Bachelor of Science Dissertation

by

Richard Gibsonborn on the 4th of January 1989

August 8, 2012

Supervisor:Dr. C. Hughes

University of York

1

Acknowledgements: I would like to thank my supervisor Dr. Christo-pher Hughes for his time and the enthusiasm he has given by meetingto discuss the project and his comments on my work as well as hisinitial explanations of some of the areas I would be researching.

Contents

1. The Riemann-zeta function 31.1. The Riemann-zeta function 31.2. The Riemann Hypothesis 51.3. The location and number of zeros of the Riemann-zeta

function 71.4. The Lindelof Hypothesis 101.5. The Riemann-Siegel formula 111.6. The Riemann-Siegel Integral Formula 112. The Dirichlet L function 132.1. The Dirichlet L function 132.2. Landau’s theorem 162.3. The Generalised Riemann Hypothesis 182.4. Multiplying series and the Functional Equation 182.5. Hurwitz-zeta function 192.6. The General Dirichlet series and Bohr functions 203. Modular forms 213.1. Modular forms 213.2. Zeros of an entire modular form 223.3. The Eisenstein series 233.4. Hecke operators 243.5. The connection between Modular forms and Dirichlet

series 254. Elliptic curves 274.1. Elliptic curves 274.2. The Riemann Hypothesis for Elliptic curves over a finite

field 284.3. The functional equation 294.4. Convergence of the Elliptic curve 295. Selberg Class 305.1. Definition of a Selberg Class 305.2. Degree of the Selberg class 315.3. Selberg’s conjectures 325.4. Poles of the Selberg Class 335.5. Modular forms in the Selberg Class and the Sato-Tate

conjecture 34

2

5.6. Rankin-Selberg convolution 355.7. Examples involving Selberg Classes 355.8. Functions in the Selberg Class 37References 39

Introduction: This project is devoted to the Selberg Class. Insections 1-4 we will explore different types of Selberg Class. Section 1will be devoted to the Riemann-zeta function, section 2 to the DirichletL-functions, section 3 to Modular forms and section 4 to Elliptic curves.Then in section 5 we will define and explore the Selberg Class and someof Selberg’s conjectures regarding the Selberg Class before I concludemy findings in section 6.

The Selberg Class.

1. The Riemann-zeta function

Figure 1. Bernhard Riemann.

1.1. The Riemann-zeta function. The Riemann-zeta function isnamed after German Mathematician Bernhard Riemann(1826-66). Itis widely used and of great importance in the field of Number Theoryand also has applications in other areas such as Probability Theory,Physics and Statistics (eg. the Casimir effect in Physics).

The Riemann-zeta Function is important due to its relation to thedistribution of prime numbers and the Prime Number Theorem andhas to be understood to understand the Riemann Hypothesis.

Definition 1. The Riemann-zeta function denoted by ζ(s) is a functiondefined over the complex plane for one complex variable, denoted s, suchthat the real part of s is greater than one.

ζ(s) =∞∑n=1

1

ns= 1 +

1

2s+

1

3s+ . . .

3

4

Properties of the Riemann-zeta function include, if for any ε >0, Re(s) ≥ 1 + ε then

∞∑n=1

1

ns

is absolutely and uniformly convergent. Therefore, all ζ(s) are abso-lutely convergent as Re(s) > 1.

The connection between the Riemann-zeta function and the primenumbers was first denoted by Swiss Mathematician Leonard Euler(1707-83). He proved that

ζ(s) =∞∑n=1

1

ns=∏p

1

1− p−s.

We will now prove this.

Proof.

ζ(s) = 1 +1

2s+

1

3s+ . . . =

∞∑n=1

1

ns.

We now use an ancient algorithm created by Greek MathematicianEratosthenes of Cyrene (276 -195.BC.) called the Sieve of Eratosthenes.The algorithm works efficiently for all primes up to the value of 10million.

First, we know that

1

2sζ(s) =

1

2s+

1

4s+

1

6s+ . . . .

We now subtract this from ζ(s).

(1− 1

2s)ζ(s) = 1 +

1

3s+

1

5s+

1

7s+ . . . .

This removes all even values of n from the equation which of course arenot prime (with the exception of 2) as they are multiples of 2. Now werepeat for the next term which has not been crossed out, this term is3.

1

3s(1− 1

2s)ζ(s) =

1

3s+

1

9s+

1

15s+ . . . .

We subtract this from (1− 12s

)ζ(s).

(1− 1

3s)(1− 1

2s)ζ(s) = 1 +

1

5s+

1

7s+

1

11s+ . . . .

As Re(s) > 1 this converges so repeating infinitely we get

. . . (1− 1

5s)(1− 1

3s)(1− 1

2s)ζ(s) = 1.

5

Therefore, if we divide both sides by . . . (1− 15s

)(1− 13s

)(1− 12s

) we get

ζ(s) =1

(1− 12s

)(1− 13s

)(1− 15s

) + . . .=∏p

1

1− p−s.

�

Using this result it is easy to show the Riemann-Zeta function isboth non-zero and analytic for all s provided Re(s) > 1.

Riemann was able to extend the domain of this definition of ζ to thewhole plane. In 1859, Riemann showed that the Riemann-zeta functionsatisfies the functional equation

ζ(s) = 2sπs−1Γ(1− s) sin(πs

2).

For alls ∈ C except the values 0 and 1, where Γ(s) =∫∞

0e−xxs−1ds is

the gamma function for s.

Using the functional equation we are able to see the Riemann-zetafunction had trivial zeros at s = −2,−4,−6 . . .. They are trivial astheir existence is easy to prove as for s = −2,−4,−6 . . ., sin(πs

2) = 0

implying the Riemann-zeta function equals to 0 from the earlier for-mula.

1.2. The Riemann Hypothesis. The Riemann Hypothesis is aboutthe distribution of the zeros of the Riemann-zeta function. It is oneof the Clay Mathematics Institutes Millennium Prize Problems andits solving is worth a million dollars and although remains unsolved ithas been shown by computer calculations that it holds for the first 10trillion zeros.

The Riemann Hypothesis states that all zeros of the Riemann-zetafunction ζ(s) in the region 0 ≤ Re(s) ≤ 1 , called the critical strip, lieon the line s = 1

2+ it.

On Riemann’s 1859 paper named “On the number of primes lessthan a given magnitude”, Riemann’s principle result was

J(x) = Li(x)−∑

Im(p)>0

[Li(xp)]− log(2) +

∫ ∞x

dt

t(t2 − 1) log(t)

where J(x) = x! whenever x! is a natural number. This denotes thenumber of primes where pn counts as 1

nof a prime and Li(x) is the

logarithmic integral function, given by∫ x

2dt

log +t.

6

But Riemann was not after a formula for J(x), he required one forπ(x) which is the number of primes less than any given magnitude, x.Because the number of primes squares, p2 that are less than x equalsto the number of primes, p less than x

12 , denoted as π(x

12 ), the number

of prime nth powers, pn less than x equals to π(x1n ), we can calculate

the formula

J(x) = π(x) +1

2π(x

12 ) +

1

3π(x

13 ) + . . . .

Using the inverse Mobius formula

f(n) =∑d|n

g(d)⇐⇒∑d|n

f(d)µ(n

d)

we get

π(x) = Π(x)− 1

2Π(x

12 )− 1

3Π(x

13 )− 1

5Π(x

15 ) +

1

6Π(x

16 )− 1

7Π(x

17 ) + . . .

=∞∑n=1

µ(n)

nΠ(x

1n )

where ∑n|N

µ(n) =

{1 N = 1,

−1 o.w..

When we substitute in the original formula for π(x)

J(x) = Li(x)−∑

Im(p)>0

[Li(xp)]− log(2) +

∫ ∞x

dt

t(t2 − 1) log(t)

into this we get

π(x) ≈ Li(x)− 1

2Li(x

12 )− 1

3Li(x

13 )− 1

5Li(x

15 )+

1

6Li(x

16 )− 1

7Li(x

17 )+ . . .

as we can ignore the terms which do not grow as x grows and periodicterms.

This implies that π(x) ∼ Li(x) is Gauss’s approximation, whereasRiemann’s approximation is

π(x) ∼ Li(x) +∞∑n=2

µ(n)

nLi(x

1n )

which is a more accurate approximation than Gauss’s approximation.

Here is a diagram of the distribution of the Riemann-zeta function.

7

The Riemann-zeta function is holomorphic at every point apart froms = 1 so is therefore a meromorphic function with a simple pole ats = 1.

1.3. The location and number of zeros of the Riemann-zetafunction. To make the locating of zeros simpler Riemann created an-other function named the Xi-function. This is defined to have the samenon-trivial zeros as the Riemann-zeta function but ξ(s) is a real numberand therefore easier to locate. The Xi-function is defined as

ξ(s) =1

2s(s− 1)

γ( s2)

πs2

ζ(s).

This also satisfies the functional equation ξ(s) = ξ(1 − s) much inthe same way as the Riemann-zeta function does.

The number of non-trivial zeros in the critical strip above the realaxis with imaginary part less than or equal to T is given by the formula

N(T ) = |(p = β + iγ|ζ(p) = 0, 0 < β < 1, 0 < γ ≤ T )| .

It is also possible to rewrite this formula.

First let f(s) be analytic inside on λ.

The formula for the number of zeros of f(s) inside λ is

1

2πi

∫λ

f ′

f(s)ds =

1

2πi∆λ log f(s) +

1

2π∆λ arg f(s).

If we apply this to the functional equation and multiply by s(s− 1) toget rid of the poles of ζ(s) and Γ( s

2) at the points s = 1 and s = 0 we

get the formula

N(T ) =1

2π∆λ arg ζ(s) =

1

π∆λ2 arg ζ(s).

8

If f is a product then f ′

fis a sum of individual pieces

∆λ2 arg(s− 1) =π

2+ 0(

1

T)

and

∆λ2π− s

2 = −T2

log(π).

Therefore,

∆λ2 arg Γ(s

2+ 1)

=1

2T log

(T

2π+

3π

8

)+ 0

(1

T

).

Applying this to the formula of N(T ) we get the equation

N(T ) =T

2πlog

(T

2πe

)+

7

8+ 0

(1

T

)where S(T ) = 1

πarg ζ

(12

+ iT).

In 1914, Hardy proved in his paper “Sur les zeros de la fonctionζ(s) de Riemann” that the Xi-function has infinitely many roots andtherefore infinitely many zeros. He proved that if N0(T ) represents thenumber of zeros of the form 1

2+ γt where 0 < γ ≤ T then N0(T )→∞

as T →∞. Hardy later, together with Littlewood extended this to thenumber of real roots between 0 and T is at least kT for some positiveconstant k and sufficiently large T.

Norwegian mathematician Atle Selberg(1917-2007) later proved inhis paper “An elementary proof of the prime number theorem” thatthe number of real roots between 0 and T to be at least kT log T wherek is a small number.

In 1974, American mathematician Norman Levinson(1912-1975) provedon his paper “more than one thirds of the zeros of the Riemann-zetafunction are on σ = 1

2” that the number of zeros of Riemann-zeta func-

tions that lie on the critical line is greater than a third of the numberof zeros of Riemann-zeta functions.

The proof of this would take over 50 pages of working and althoughusing many of Selbergs ideas, differs greatly from Selbergs work. Theproof was done by relating the Riemann-zeta functions zeros to itsderivative.

In 1989, American mathematician John Brian Conrey further im-proved this estimate to 40 percent on his paper “more than two fifthsof the zeros of Riemann-zeta function are on the critical line.” Thiswas further improved to 41 percent in 2010 by Conrey and Americanmathematicians Matthew Young and Hung Biu.

9

But even the statement that more than 41 percent of zeros of aRiemann-zeta function lie on the critical line σ = 1

2is a long way off

proving the Riemann Hypothesis that all zeros of the Riemann-zetafunction lie on the critical line.

From earlier we know that the only zeros of the Riemann-zeta func-tion below the line Re(s) < 0 are the trivial values. Using the factthat the pole of the Riemann-zeta function is 1 and for Re(s) > 1 theequation ζ(s) =

∑p

11−p−s the factors of which are all non-zero and

holomorphic which contradicts that ζ(s) has a pole at s = 1. Thisimplies that ζ(s) has no zeros for Re(s) > 1.

Therefore, all non-trivial zeros of the Riemann-zeta function lie inthe critical strip, 0 ≤ Re(s) ≤ 1.

In 1896 French mathematician Jacques Hadamard(1865-1963) andBelgian mathematician Charles Jean de la Vallee-Poussin(1866-1962)proved that the upper boundary of the critical strip contained no zeros.Hadamards proof was the simpler of the two and can be shown by

log ζ(s) =

∫ ∞0

x−sdJ(x) =∑p

1

ps+

1

2

∑p

1

p2s

1

3

∑p

1

p3s+ . . . .

If we set

B =1

2

∑p

1

p2+

1

3

∑p

1

p3+ . . .

we get the equation

Re(log ζ(s)) ≥∑p

cos(t log p)

pσ−B

This value can approach −∞ as σ → 1 only if the first term approaches−∞.

This implies that

limn→1

∑p

cos(t log p)

pσ= −∞

which is impossible and therefore Re(s) 6= 1 for any zeros of theRiemann-zeta function.

10

1.4. The Lindelof Hypothesis. The Riemann Hypothesis was notthe only important hypothesis involving the Riemann-zeta function.The Lindelof Hypothesis created by Finnish mathematician Ernst LeonardLindelof(1870-1946) although not as strong as the Riemann Hypothe-sis as the Riemann Hypothesis implies the Lindelof Hypothesis but notvica versa. Lindelof stated that for every positive ε, ζ(1

2+ it) = 0(tε)

where 0 is the “big Oh” notation.

We say that f(x) = 0(g(x)) as x → ∞ if there exists a constant Kand a value x0 such that |f(x) < Kg(x)| whenever x ≥ x0.

This is a conjecture on the maximum size of the Riemann-zeta func-tion and can be better understood looking at the 2kth moment of theRiemann-zeta functions modulus. This was first done by British math-ematicians John Edensor Littlewood(1885-1977) and Godfrey HaroldHardy(1877-1947).

The Lindelof Hypothesis can also be described as for any k and anyε > 0, Ik(T ) = (T ε) The 2kth moment is defined as

Ik(T ) =1

T

∫ T

o

∣∣∣∣∣ζ(

1

2+ it

)2k∣∣∣∣∣ dt.

The moments represent the Riemann-zeta functions mean values onthe line Re(s) = 1

2. This is important as these means let us understand

the zeros of the Riemann-zeta function in greater detail.

In 1918 Hardy and Littlewood discovered the average value of |ζ(s)|2where Re(s) = 1

2is infinite and I1(T ) ∼ log T and in 1922 further

simplified this proof by using the approximate functional equation forζ(s). Since then the 4th, 6th and 8th moments have been calculatedas

I2(T ) = 12π2 log4 T = 0(log3 T ) in 1926 by British mathematician

Albert Edward Ingham(1900-67),

I3(T ) ∼ 429!a(3)(log T )9

and

I4(T ) ∼ 2402416!

a(4)(log T )16

which were both conjectured by John Brian Conrey and anotherAmerican mathematician Steve Gonek.

However, there is no proof of the Lindelof hypothesis as the momentsfor k > 4 have not been found.

11

In 1918, Swedish mathematician Ralf Backlund showed the LindelofHypothesis is equivalent to the statement that for every ε > 1

2the

number of roots in the rectangle (T ≤ Im(s) ≤ T + 1;σ ≤ Re(s) ≤ 1)grows less rapidly than log(T ) as T →∞.

This is also equivalent to saying that for every ε > 0 there exists aT0 such that the number of zeros is less than ε log T whenever T ≥ T0.

If the Riemann Hypothesis is true there are no zeros in (T ≤ Im(s) ≤T + 1;σ ≤ Re(s) ≤ 1) which implies the Lindelof Hypothesis is true ifthe Riemann Hypothesis is true.

1.5. The Riemann-Siegel formula. The Riemann-Siegel formula wasfound by German mathematician Carl Ludwig Siegel(1896-1981) inRiemann’s unpublished papers in the archives of Gottingen’s Univer-sity Library. In these papers Siegel found the Riemann-Siegel formulawhich is for Z(t) and also a way to represent ζ(s) in terms of definiteintegrals.

The Riemann-Siegel formula is a formula estimating the error of theapproximate functional equation of the Riemann-zeta function.

The Z-function is used to study the Riemann-zeta function alongthe critical line. It has exactly the same zeros as the Riemann-zetafunction along the critical line and is defined as

Z(t) = eiθ(t)ζ

(1

2+ it

).

The formula states that

Z(t) = 2R

(eiθ(t)

(∞∑n=1

Q(s

2, πin2

)− π

s2 e

πis4

sγ( s2)

)).

The density of these Z-functions on the critical line is k2π

log(ktπ

)where k is a constant and k > 0.41 due to the findings stated earlierthat more than 41 percent of zeros lie on the critical line. For theRiemann Hypothesis to be true k must equal 1.

1.6. The Riemann-Siegel Integral Formula. Earlier in 1926, Ger-man mathematician Bessel-Hagen(1898-1946) found a representation ofthe Riemann-zeta function in terms of definite integrals in Riemann’spapers. Siegel included this formula in his paper and stated the formula

2ζ(s)

s(s− 1)= F (s) + F (1− s)

12

where

F = J(s

2− 1)π−

s2

∫0↘1

e−iπx2x−sdx

eiπx − e−iπx.

0 ↘ 1 means the path of integration is a line of gradient -1 whichcrosses the real axis between 0 and 1.

Therefore, we can see F is an analytic formula defined for all s exceptpossibly at the points s = 0,−2,−4, . . . where J( s

2−1) has simple poles.

But using the Riemann-Siegel integral formula it is shown that F (s)is analytic at −2,−4, . . . but has a simple pole at 0.

13

2. The Dirichlet L function

2.1. The Dirichlet L function. The Dirichlet L-functions are namedafter Johann Peter Gustav Lejeune Dirichlet(1805-59) who introducedthem in 1837 as part of the proof for the theorem of primes in arith-metic progressions.

A Dirichlet L-function is defined as

L(s, χ) =∞∑n=1

χ(n)

ns

where s is a complex variable with Re(s) > 1 and χ is a Dirichletcharacter modulo m and

χ(n) =

{1 if g.c.d(n,m) = 1

0 if g.c.d(n,m) 6= 1.

If this function is holomorphic on all of an open subset except at thepoles for the whole complex plane then L(s, χ) is a Dirichlet L-function.

Theorem 1. Dirichlets theorem of primes in arithmetic progressionstates that for any two positive coprime integers a and d, there existinfinitely many primes of the form a+ nd, where n ≥ 0.

This is an extension of the theorem of Ancient Greek mathemati-cian Euclid that there are infinitely many prime numbers. Dirichletstheorem was first conjectured by French mathematician Adrien-MareLegendre(1752-1833) in 1831 in his unsuccessful attempts to solve prob-lems of quadratic equations modulo prime numbers.

14

Theorem 2. If m ∈ Z+ and χ is a Dirichlet character modulo m,and (a,m) = 1 then there exist infinitely many primes of the formp ≡ amodm.

Proof. To prove this we need to show that∑p≡amodm

=1

p

diverges.logL(s, χ) can be defined on the half plane Re(s) > 1 by

logL(s, χ) =∑p,k

χ(pk)

kpks

where p is prime and k is a positive integer.This can be rewritten as

logL(s, χ) =∑p

χ(p)p−s +∑p

∞∑k=2

χ(pk)

kpks.

If we set

R(s, χ) =∑p

∞∑k=2

χ(pk)

kpks

which clearly converges for Re(s) > 12.

Also, if (a,m) = 1, there must be an integer b such that ab ≡ 1modm.

We can see that∑χ

χ(bp) =

{h if bp ≡ 1modm

0 o.w.

So if we multiply both sides by∑

χ χ(b) we get∑χ

χ(b) logL(s, χ) =∑p

∑χ

χ(bp)p−s +∑χ

χ(b)R(s, χ)

= h∑

p≡amodm

p−s +∑χ

χ(b)R(s, χ).

As R(s, χ) converges for Re(s) > 12

so does∑χ

χ(b)R(s, χ).

If we note that logL(s, χ) can also be written as

logL(s, χ) = −∫ c

s

L′(u, χ)

L(u, χ)du+ logL(c, χ)

15

for s = Re(s) > 1, c > Re(s) which diverges.

This implies the right hand side of the equation diverges also.

Therefore, as ∑χ

χ(b)R(s, χ)

converges

h∑

p≡amodm

p−s

diverges.

As h is a constant we can see that∑p≡amodm

p−s

diverges which is what we set out to prove. �

As part of his proof Dirichlet showed the L-function, L(s, χ) to benon-zero at s=1 implying if χ is principal then L(s, χ) has a simplepole at s = 1.

We can tell the Dirichlet L-function

L(s, χ) =∞∑n=1

χ(n)

ns

converges for Re(s) > 1 as because

|χ(n)| ≤ 1

it can be compared to∞∑n=1

1

ns

which converges for Re(s).1.

Therefore L(s, χ) converges for Re(s) > 1.

Much in the same way as the Riemann-zeta function, the DirichletL-function has negative trivial value for its real part, although theydepend on the value of χ and only exist if χ is a primitive character.

For Re(s) < 0, if χ(−1) = 1 then the only zeros of L(s, χ) whereRe(s) < 0 are the negative even integers, and if χ(−1) = −1 then thezeros of L(s, χ), where Re(s) < 0 are the negative odd integers.

The Dirichlet L-function also has an Euler product which is similarto the Euler product of the Riemann-zeta function.

16

The Euler product for the Riemann-zeta function is

ζ(s) =∏p

(1− p−s)−1

which can also be written as

ζ(s) =∑p

(ζp(s))

where

ζp(s) =∞∑j=0

1

pjs.

The Euler product for a Dirichlet L-function is also of the form

L(s, χ) =∏p

Lp(s)

which is very similar to that of the Riemann-zeta function.

We have

Lp(s) =∞∑j=0

apj

pjs

which is the same as ζp(s) with every value of the sum multiplied by apower of ap.

The Riemann Hypothesis might not hold for Dirichlet L-functionsdue to the possible existence of a Siegel zero. The Siegel zero is namedafter German mathematician Carl Ludwig Siegel(1896-1981) and is acounter-example of the Riemann Hypothesis for the zeros of DirichletL-functions.

There are hypothetical complex values, s very close to 1 where L(s, χ) =0 for a Dirichlet character, χ to the modulus q. Therefore, not allDirichlet L-functions values have real part equal to 1

2, although zero

free regions past the line Re(s) = 1 exist much in the same way as forthe Riemann-zeta function.

17

2.2. Landau’s theorem. In 1953 H. G. Landau proved in his paper“On dominance relations and the structure of animal societies” anothertheorem regarding the Dirichlet series.

Theorem 3. If a0 ≥ 0 for all n ≥ 1 and σ0 is finite, then the point ofintersection of the real axis with the line of convergence is a singularityof the sum function f(s) of the Dirichlet series

∞∑m=1

anns.

Proof. Without loss of generality we may assume that σ0 = 0.

Now to prove this theorem we must show that s = 0 is a singularityof f .

The Taylor series of f at s = 1 would have radius of convergenceρ > 1 and the Taylor series could be written as

∞∑v=0

(s− 1)v

v!f (v)(1).

There would also exist a real s < 0 for which this Taylor series con-verges.

We also know that

f(s) =∞∑n=1

ane−s logn

for all σ > 0.

We now need f (v)(1).

First we derive

f (v)(s) =∞∑n=1

an(− log n)v

ns

and then take s = 1 to get

f (v)(1) =∞∑n=1

an(− log n)v

n.

18

Substituting this into the Taylor series of f at s = 1 gives

∞∑v=0

(s− 1)v

v!

∞∑n=1

an(− log n)v

n

=∞∑v=0

(1− s)v

v!

∞∑n=1

an(log n)v

n

=∞∑n=1

ann

∞∑v=0

(1− s)v(log n)v

v!

=∞∑n=1

anne(1−s) logn

which must be convergent for some s < 0.

But∞∑n=1

ann

must diverge which gives a contradiction.

Therefore s = 0 is a singularity of f(s). �

2.3. The Generalised Riemann Hypothesis. For the Dirichlet L-function a new hypothesis replaces the Riemann Hypothesis. This iscalled the Generalised Riemann Hypothesis. The Generalised RiemannHypothesis states that a Dirichlet character is a completely arithmeticfunction, χ such that there exists a positive integer, k with χ(n+ k) =χ(n) for all n and χ(n) = 0 whenever g.c.d(n, k) > 1. If such a characteris given we define the corresponding Dirichlet L-function by

L(s, χ) =∑∞

n=1χ(n)ns

for every complex number, s with Re(s) > 1.

Then for every Dirichlet character, χ and every complex number, swith L(s, χ) = 0; if 0 < Re(s) < 1 then Re(s) = 1

2.

2.4. Multiplying series and the Functional Equation. If we takeany two convergent Dirichlet series f and g defined as

f(s) =∞∑n=1

akks

g(s) =∞∑n=1

bnns

19

then the Dirichlet product h is defined as

h(s) = f(s)g(s) =

(∞∑k=1

akks

)(∑ ∞∑n=1

anns

)=∞∑k=1

∑n|k

anb kn

1

ks.

The Dirichlet L-functions also have a functional equation. It can bedefined as

L(s, χ) = εL(1− s, χ∗)where χ is a Dirichlet character and χ∗ its complex conjugate. ε is alsoa complex value with absolute value of 1. In the case where χ ∈ R thenboth sides of the equation are equal when ε = 1.

However, a stronger result can be obtained using the Hurwitz-zetafunction.

2.5. Hurwitz-zeta function. The Hurwitz-zeta function was intro-duced by German mathematician Adolf Hurwitz(1859-1919) and canbe used to show similar properties of the Riemann-zeta function andthe Dirichlet L-function.

Definition 2. The Hurwitz-zeta function ζ(s, a) is defined by

ζ(s, a) =∞∑n=0

1

(n+ a)s

for σ > 1 where a is a fixed real number, 0 < a ≤ 1.

When a = 1 we get ζ(s, 1) = ζ(s) which is the Riemann-zeta func-tion. It is also possible to express the Dirichlet L-function in terms ofHurwitz-zeta functions.

To do this we take n = qk + r, with 1 ≤ r ≤ k and q = 0, 1, 2, . . .This gives

L(s, χ) =∞∑n=1

χ(n)

ns=

k∑r=1

∞∑q=0

χ(qk + r)

(qk + r)s

=1

ks

k∑r=1

χ(r)∞∑q=0

1

(q + rk)s

= k−sk∑r=1

χ(r)ζ(s,r

k

).

From these results we can see the properties of both the Riemann-zeta function and the Dirichlet L-function both depend on the Hurwitz-zeta function.

20

2.6. The General Dirichlet series and Bohr functions. A generalDirichlet series is an infinite series defined as

f(s) =∞∑n=1

χ(s)e−sλ(n)

where χ(s) and s are complex numbers and λ(n) is a strictly increasingsequence of positive numbers which tend to ∞.

The Dirichlet series can be defined from f(s) by replacing λ(n) bylog n and the kth derivative of f(s) can be denoted by

f (k)(s) = (1)k∞∑n=1

χ(n)λk(n)e−sλ(n).

To study the Bohr functions we must first define Bohr matrices andbases.

Definition 3. An infinite square matrix R = (rij) with rational entriesis a Bohr matrix if all but a finite number of entries in each row equalzero.

Definition 4. B is a basis for a Dirichlet L-function L(s, χ) if(i) RB = 0 implies R = 0(ii) There exists a Bohr matrix R such that L(s, χ) = RB(iii) There exists a Bohr matrix T such that B = TL(s, χ).

Now we have all the information needed to define a Bohr function.

Definition 5. For every General Dirichlet series

L(s, χ) =∞∑n=1

χ(s)e−sλn

there exists an associated Bohr function F (z1, z2, . . .) of countable manycomplex variables z1, z2, . . ..

The Bohr function is defined as

F (z1, z2, . . .) =∞∑n=1

χ(s)e−(R(z1,z2,...))−n.

Now if we replace (z1, z2, . . .) with Z we get the equation

F (Z) =∞∑n=1

χ(s)e−(RZ)n

where (RZ)n represents the nth entry of column matrix RZ.

21

3. Modular forms

3.1. Modular forms. A modular form is an analytic function on theplane with positive imaginary part (H = x+ iy|y > 0;x, yεR) whichsatisfies a certain kind of functional equation and growth condition.

Definition 6. A function f is an entire modular form of weight k if

(i) f is analytic in H = x+ iy|y > 0;x, yεR.(ii) f(aτ+b

cτ+d) = (cτ+d)kf(τ) (the functional equation of a modular form)

whenever

[a bc d

]is a member of the modular group gamma.

(iii) The Fourier Series of f has the form

f(τ) =∞∑n=0

c(n)e2πinτ

Modular forms do not just appear in Number Theory but also inother areas of mathematics such as Complex Analysis and Topology.

Using condition (iii) we can show that an entire modulo form isanalytic at the point i∞ and everywhere on the half plane, H.

We can see this as condition (iii) states the Laurent expansion of anentire modular form contains no negative powers of x and the Fourierexpansion of a function of period 1 is equivalent to its Laurent expan-sion near x = 0 and x = e2πinτ .

For x to equal 0 we need either f to either be on the half plane, Hor for f to be i∞.

22

The value c(0) is called the cusp form and is the value of f at i∞.At this value no Eisenstein series G2k vanishes with G2k denoted by

G2k =∑

(m,n)6=(0,0)

1

(m+ nτ)2k

where k is an integer greater than or equal to 2.

A modular form which is not entire may have poles at the points i∞and at values along the half plane, H.

3.2. Zeros of an entire modular form. Define the open set

RΓ = {τ ∈ H : ‖τ‖ > 1, ‖τ + τ‖ < 1}as a fundamental region for Γ.

Then an entire modular form which does not equal 0 with N zeros inthe closure of RΓ has the weight k = 12N +6N(i)+4N(ρ)+12N(i∞).

We can prove this as we know f has no poles, so we can write

N =1

2πi

∫∂R

f ′(τ)

f(τ)dτ

where the path ∂R is the path along the edges of R avoiding the zerosat the vertices and any other zeros occurring on the edges.

If we then calculate the limit of the integral we get the formula

N =k

12− 1

2N(i)− 1

3N(ρ)−N(i∞)

and by dividing by 12 and addition and subtraction of values to eachside of the equation we get

k = 12N + 6N(i) + 4N(ρ) + 12N(i∞)

which is what we set out to prove. We call this formula the weightformula.

Using this formula we can study the entire modular forms for certainvalues of k. Firstly, the only entire modular forms of weight k = 0 arethe constant functions. We can show this as any entire modular formwith weight, k = 0 is a modular function defined as a function f whichsatisfies any of the 3 properties for an entire modular form. Also as itis analytic at every point in the half plane it must be constant.

By studying the weight equation we can also see that if k is a positiveodd integer or k = 2 then the only entire modular form of weight k isthe zero function. This can be shown as N , N(i), N(ρ) and N(i∞)are all non-negative therefore using the equation for k we can see that

23

k is even and cannot be negative. Therefore, the only entire modularform of weight k in this case is the zero function. Further extendingthis work we can show that k ≥ 4 if k is odd.

Furthermore, we can show the only entire cusp form of weight k < 0is the zero function as if k < 12 then N(i∞) = 0 and therefore if f isa cusp then it must equal 0.

3.3. The Eisenstein series. For the Eisenstein series of order k fork > 2 defined as

Gn =∑w 6=0

1

wn

we have the invariants g2 = 60G4 and g3 = 140G6 such that ∆ is thediscriminant and is a polynomial in G4 and G6.

We define the discriminant ∆ = g23−27g3

2 = (60G4)3−27(140G6)2.

To show every entire modular form with weight, k > 2 is a poly-nomial in G4 and G6 we take two entire modular forms f and g withweights k1 and k2. We know the product fg is an entire modular formwith weight k1 + k2 and f

gis also an entire modular form with weight

k1 − k2 if g has no zeros in H or at the point i∞.

It is also possible to express any entire modular form in terms ofG4, G6,∆ and Klein’s modular invariant J .

Klein’s modular function is homogeneous of degree 0 and is a com-bination of g2 and g3.

Definition 7. If k2k1

is not real we define

J(τ) = J(k1, k2) =g2

3(k1, k2)

∆(k1, k2).

Therefore, if f is an entire modular form with N zeros z1, z2, . . . , zN inthe closure of RΓ then there exists a constant c such that

f(τ) = cG4(τ)N(ρ)G6(τ)N(i)∆(τ)N(i∞)∆(τ)NN∑k=1

(J(τ)− J(zk).

Proof. First look at the function

g(τ) =N∑k=1

(J(τ)− J(zk).

24

This function is an entire modular form with its only zeros in the closureof RΓ. These are at the points z1, z2, . . . , zN and with a pole of orderN at i∞.

We also know that the discriminant ∆ has a zero of first order atthe point i∞ and ∆Ng is an entire modular form in the closure of RΓ,vanishing at the points z1, z2, . . . , zN .

The weight of ∆Ng is 12N .

Manipulating this we can see that a function h defined as

h = G4N(ρ)G6

N(i)∆N(i∞)∆Ng

has identical zeros to the function f in the closure of RΓ.

We can see the weight of h is

k = 4N(ρ) + 6N(i) + 12N(i∞) + 12N.

This is the same as the weight for f and implies h is an entire modularform with weight identical to that of f. Therefore f

his an entire modular

form of weight 0 which implies fh

is constant and therefore proves theformula for f(τ) to be correct. �

3.4. Hecke operators. Hecke operators introduced by German math-ematician Erich Hecke(1887-1947)to help determine all entire modularforms with multiplicative coefficients. The Hecke operators are denotedby Tn where n = 1, 2, . . . and they map Mk onto itself.

Mk is a linear space over a complex field where Mk is the set of allentire modular forms of weight k.

Definition 8. For a fixed integer k and any n ∈ (1, 2, . . .) Tn is definedon Mk as

(Tnf)(τ) = nk−1∑d|n

d−kd−1∑b=0

f(nτ + bd

d2).

Proof. To prove that Tn maps each entire modular function , f in Mk

onto another function in Mk we must first use the Fourier expansion.

f has the Fourier expansion

f(τ) =∞∑m=0

e2πimτ .

25

If we apply the definition of Tn to the Fourier series we get the equation

(Tnf)(τ) = nk−1∑d|n

d−kd−1∑b=0

∞∑m=0

c(m)e2πim(nτ+bd)

d2

=∞∑m=0

∑d|n

(n

d)k−1c(m)e

2πimnτd2

1

d

d−1∑b=0

e2πimbd .

If we set m = qd then using the fact that the sum on b equals to d ifd|m and is zero otherwise we can get the equation

(Tnf)(τ) =∞∑q=0

∑d|n

(n

d)k−1c(qd)e

2πimbd .

Then replacing d by nd

and q by md

we get

(Tnf)(τ) =∞∑q=0

∑d|nd|m

(d)k−1c(mn

d2)e2πimτ .

The d|m occurs as m = qd.

So if f ∈ Mk has a Fourier expansion then Tnf has the Fourierexpansion we just calculated. �

It is also possible to write the equation for Hecke transformations as

(Tnf)(τ) = nk−1∑

a≤1,ad=n0≤b<d

d−kf(Aτ)

=1

n

∑a≤1,ad=n

0≤b<d

akf(Aτ)

where Aτ = aτ+bd

and n = ad.

This can further be simplified into the form

(Tnf)(τ) =1

n

∑A

akf(Aτ)

where matrix A =

[a b0 d

]and d runs through the positive divisors of n

and b runs through a complete residue system modulo d. Also, a = ndand is the first value of the first column of matrix A.

26

3.5. The connection between Modular forms and Dirichlet se-ries. If we take any modular form with Fourier series

f(τ) = c(0) +∞∑n=1

c(n)e2πinτ

and a Dirichlet series of the form

φ(s) =∞∑n=1

c(n)

ns

then they will both be formed with the same coefficients except forc(0).

Hecke also proved that if f ∈M2k then the order of c(n) is nk is f isa cusp form or n2k−1 is f is not a cusp form.

This information proves the convergence of the Dirichlet series φ(s).φ(s) converges absolutely for Re(s) > k + 1 in the case where f is acusp form and for Re(s) > 2k when f is not a cusp form.

27

4. Elliptic curves

4.1. Elliptic curves.

Definition 9. A function f is called elliptic if

(i) f is doubly periodic, i.e. f is defined at all points on the complexplane and has two complex numbers u and v which are named periodswhich are linearly independent as vectors over the field of real numbersand

f(z) = f(z + u) = f(z + v).

(ii) f is meromorphic, i.e. its only singularities are poles.

There exists an L-function on an elliptic curve which was first intro-duced by German mathematician Helmut Hasse(1898-1979) and laterextended by French mathematician Andre Weil(1906-1998). It is there-fore named the Hasse-Weil zeta function. There are two properties of

28

the Hasse-Weil zeta function, the first states that an L-function de-fined a Dirichlet series for Re(s) > 3

2has an analytic continuation to

the complex plane and satisfies a functional equation under reflexion sto 2-s.

The second property was conjectured by British mathematicians Pe-ter Swinnerton-Dyer(1927-present) and Bryan John Birch(1931- present).It is known as the Birch and Swinnerton-Dyer conjecture and is anotherof the Clay Mathematics Institutes Millennium Problems.

The First Birch and Swinnerton-Dyer conjecture: Let E be an ellipticcurve over a number field K with LE|K(s) having analytic continuationto the complex plane. Let gE|K denote the rank of E(K). Then gE|Kis equal to the order zero of LE|K(s) at the point s = 1.

This conjecture has only been proven for rank less than or equal toone but a weaker form of it has been. It was created by Australianmathematician John H. Coates(1945-present) and states that

ords=1LE|K(s) ≥ gE|K

iff IIIE|K is finite.

Coates along with British mathematician Sir Andrew John Wiles(1953-present), who is famous for proving Fermat’s Last Theorem, also cre-ated another conjecture, this time for elliptic curves with complex mul-tiplication.

Theorem 4. Let E be an elliptic curve over the field K which is eitherthe rational numbers or the field of complex multiplication.

If gE|K 6= 0, then LE|K(1) = 0.

In other words the L-functions of E over K has a zero at s = 1.

It is also possible to have modular elliptic curves. They were shownto be elliptic curves over the rational numbers with complex multi-plication by Japanese mathematician Goro Shimura(1930-present) in1971.

He also showed that if E is a modular elliptic curve defined over therational numbers and its L-function has a zero at s = 1 then E(Q) isinfinite.

29

4.2. The Riemann Hypothesis for Elliptic curves over a finitefield. There is also a version of the Riemann Hypothesis for Ellipticcurves. It was first conjectured by Austrian-Armenian mathematicianEmil Artin(1898-1962) in 1930 and later proved by Hasse in 1934.

Theorem 5. Let E be an elliptic curve defined over a finite field Fqand let Nm = #E(Fqm). Then, for all m ≥ 1 we have

|1 + qm −Nm| ≤ 2qm2 .

This is the Riemann Hypothesis for Elliptic curves.

It is equivalent the fact that the roots of fE(T ) are complex conju-gates of each other where

fE(T ) = det(1− πET ) = 1− (Tr(π))T + qT 2

and is known as the characteristic polynomial of Frobenius.

The roots of fE(T ) have absolute values of 1√q

and therefore if we

introduce the zeta function ζE(s) defined as

ζE(s) =fE(q−s)

(1− q−s)(1− q1−s)=

(1− Tr(π))q−s + q1−2s

(1− q−s)(1− q1−s)

then ζE(s) has zeros on the line Re(s) = 12

which is more recognisableas a Riemann Hypothesis.

Using the equation for ζ(s) it is easy to see that the zeta functionhas poles at s = 0 and s = 1.

4.3. The functional equation. We know that

fE(q−s) = 1− (Trπ)q−s + q1−2s.

Let us now replace s with 1− s.

fE(q−(1−s)) = 1− (Trπ)q−(1−s) + q1−2(1−s)

= 1− (Trπ)qs−1 + q2s−1

= q2s−1(1− (Trπ)q−s + q1−2s)

= q2s−1fE(q−s).

Therefore, the functional equation for an elliptic curve is

ζE(1− s) = q2s−1ζE(s).

30

4.4. Convergence of the Elliptic curve. The Weierstrass σ-functionwas introduced by German mathematician Karl Weierstrass(1815-1897)to help prove the convergence of functions defining elliptic curves.

Definition 10. The Weierstrass σ-function is defined as

σ(z;L) = σ(z) =1

z2+∑

w∈L−0

[1

(z − w)2− 1

w2

].

The equation 1(z−w)2

− 1w2 can be rewritten as 2wz−z2

w2(z−w).

This behaves in the same way as 1w3 for sufficiently large |w| and as

we know that ∑w∈L−0

1

w3

converges absolutely. This implies σ(z) converges as w →∞.

We can compare this with the ζ function which we define as

ζ(z;L) = ζ(z) =1

z+∑

w∈L−0

[1

z − w+

1

w+

z

w2].

The sum 1z−w + 1

w+ z

w2 can be rewritten as z2

w2(z−w)which also behaves

in a similar way to 1w3 .

Therefore we can say the ζ function converges as w →∞.

31

5. Selberg Class

5.1. Definition of a Selberg Class. In Atle Selbergs 1992 papernamed “Old and new conjectures and results about a class of Dirichletseries” he introduced the idea of a “Selberg class” which is an axio-mated definition of a class of L-functions.

Definition 11. An absolutely convergent Dirichlet series F of the Sel-berg class S satisfies the following conditions.

(i) The Ramanujan conjecture: In the half plane Re(s) > 1,

F (s) =∞∑n=1

anns

is an absolutely convergent Dirichlet series with a1 = 1 and an � nε

for every ε > 0.

(ii) Analyticity: There exists an m ∈ N such that (s− 1)mF (s) is ananalytic function of finite order over the whole complex plane.

(iii) Functional equation: There exists a function

Φ(s) = Qs

r∑j=1

Γ(λjs+ µj)F (s)

where Q, λj > 0 and Re(µj) ≥ 0 such that

Φ(s) = wΦ(1− s)

where the absolute value of w is 1 and f(s) = f(s).

32

The degree of F is given by d = 2∑r

j=1 λj.

It is important to have the real part of µj greater than or equal tozero and some negative real part L-functions do not satisfy the RiemannHypothesis.

(iv) The Euler product: logF (s) is expressionable as a Dirichletseries as

logF (s) =∞∑n=2

bnnsλ(n)

log n

where bn � nθ for some θ < 12.

It is important not to include θ = 12

as this would violate the Rie-mann Hypothesis.

5.2. Degree of the Selberg class.

Conjecture 1. All functions in S have degrees that are integers.

In 1957, German mathematician Hans-Egon Richert(1924-1993) showedthat there are no elements in a Selberg class with degree 0 < d < 1.

This was further improved by Conrey and Indian mathematicianAmit Ghosh on their 1993 paper On the Selberg class of Dirichletseries: Small degrees.”

They proved that if a function is in the Selberg class it’s either equalto 1 or has a degree greater than or equal to 1.

Theorem 6. If F ∈ S then F = 1 or dF ≥ 1.

Proof. Suppose dF < 1.

We can say that

h(x) =P (log x)

x+K(x)

where

K(x) =∞∑n=0

(−1)nγF (n+ 1)F (n+ 1)(2πx)n

γF (−n)n!

and P is a polynomial.

We can see that h(x) is an analytic function as γF (n+1)γF (−n)n!

� n−(1−d)nAn

where A > 0 which shows K(x) is an entire function of x and thereforeH(x) is analytic.

33

But h is also entire as it has a period i and therefore has no singu-larities on the real axis.

Now we can set

h(iz) =∞∑n=1

anenz = H(z).

So if y > 0 then

ane−2πny =

∫ 1

0

H(z)e−nxdx

which if we differentiate both sides twice and then set y = 0 becomes

(2πn)2an = −∫ 1

0

H ′′(x)e−nxdx�∫ 1

0

|H ′′(x)| dx� 1.

From this we can derive an � 1n2 .

We know a Dirichlet series F (s) is absolutely convergent for Re(s) >−1 and is bounded for Re(s) > 1

2.

But as

F (1− s) =Φ(s)

γF (1− s)∼ γF (s)

γF (1− s)for Re(s) ≥ σ0 > 1 this is a contradiction.

To prove for F = 1 Conrey [9] contradicted the existence of a θ < 12

for which bn � nθ.�

5.3. Selberg’s conjectures. In Selberg’s paper he made several con-jectures for the Selberg class.

(i) Regularity of distribution: Each F ∈ S has a corresponding inte-ger nF such that ∑

p≤x

|ap|2

p= nF log log x+ 0(1).

Also if F is a primitive function then nF = 1 which gives∑p≤x

|ap|2

p= log log x+ 0(1).

(ii) Orthonormality: If we have two distinct and primitive functionsF and F ′ then nF = 1 and∑

p≤x

apa′pp

= 0(1).

34

(iii) GL(1) twists: If χ is a primitive Dirichlet character and F andF χ are both in the Selberg class then

F =k∑i=1

Fi

where Fi are primitive implies that

F χ =k∑i=1

Fiχ

where Fiχ are primitive elements of the Selberg class.

Selberg also stated that for all functions in the Selberg class theRiemann hypothesis holds, that is that all functions in the Selbergclass have real part equals to 1

2.

5.4. Poles of the Selberg Class. Suppose that F (s) ∈ S has a poleor zero at s = 1 + iα, α ∈ R.

Then ∑p≤x

app1+iα

are unbounded as x→∞.

Proof. We know F (s) ∼ C(s − (1 + iα))m as s = σ + iα → 1+ + iα,m ∈ (Z+).

Then

logF (s) ∼ m log(σ − 1).

By the Euler product condition of a Selberg Class

logF (s) =∞∑n=2

bnλ(n)

ns log n

=∞∑n=1

bnns

=∑p

apps

+ 0(1).

Therefore, ∑p

apps∼ m log(σ − 1)

as σ → 1+.

35

Now we show that ∑p

apps

= 0(1).

Let

S(x) =∑p≤x

app1+iα

.

If we assume S(x) is bounded then∑p

apps

=

∫ ∞1

x1−σdS(x)

= (σ − 1)

∫ ∞1

S(x)x−∞dx

= 0(1).

This is a contradiction, therefore∑p≤x

app1+iα

is unbounded as x→∞.�

5.5. Modular forms in the Selberg Class and the Sato-Tateconjecture. It is possible to connect modular forms to the Selbergclass by Langland’s conjectures.

Langland’s conjectures connect number theory and the representa-tion theory of certain groups. They allow L-functions which are asso-ciated with symmetric power representations to be in S.

Now we can connect modular forms to this by associating L(s) witha cusp form where

L(s) =∏p

(1− αp

ps

)−1(1− βp

ps

)−1

is in S and

Lm(s) =∏p

m∏j=0

(1−

αjpβm−jp

ps

)−1

where Lm(s) is the m-th symmetric power L-function then Lm(s) ∈ Sfor all m > 0.

In 1968, French mathematician Jean-Pierre Serre(1926-present) provedthat this fact together with the non-vanishing on the line σ = 1 implies

36

the Sato-Tate conjecture in his paper “Abelian l-adic representationsand elliptic curves.”

Definition 12. The Sato-Tate conjecture is

Np

p= 1 + 0

(1√(p)

)

as p→∞ where Np denotes the number of points on Ep.

Ep is a family of elliptic curves over a finite field with p elementsobtained from an elliptic curve E over the ration field by the process ofreduction modulo p.

5.6. Rankin-Selberg convolution. The Rankin-Selberg convolutionoriginated from Selbergs papers and was developed by British mathe-matician Alexander Rankin(1915-2001) in 1939 and Selberg in 1940.

For a modular form g and a Dirichlet character χ = χqχD whereχq and χD are Dirichlet characters modulo q and D. If we take f ∈Sk(q, χq) and g ∈ S∗(D,χD)∗ then we can define a L-function as

L(f ⊗ g, s) = L(χ, 2s)∑n≥1

λf (n)λq(n)

ns

=∏p

2∏i=1

2∏j=1

(1− αf,i(p)αg,i(p)

ps

)−1

where λ = r2 + 14

, r ∈ R and αf,1(p), αf,2(p), αg,1(p) and αg,2(p) arethe roots of

X2 + λf (p)X + χq(p) = 0

and

X2 + λg(p)X + χD(p) = 0.

This function is analytic over the whole complex plane except whenf = g. At these points it has poles at s = 0 and s = 1.

It has a functional equation of

Λ(f ⊗ g, s) = ε(f ⊗ g)Λ(f ⊗ g, 1− s).

37

5.7. Examples involving Selberg Classes.

Lemma 1. For a q-periodic function f , L(s,f)ζ(s)

is an entire function

implies that

L(1− s, f) =2√q

( q

2π

)sΓ(s) cos

(πs2

)L(s, f+).

Proof. Because ζ(s) has trivial zeros when s is a negative even integerL(s, f) must vanish there also.

Using the functional equation

L(1−s, f) =( q

2π

)s Γ(s)√q

(exp

(πis

2

)L(s, f−) + exp

(−πis

2

)L(s, f+)

)with

f±(n) =1√q

∑amodq

f(a) exp

(±2πi

an

q

)we can show

L(2m+ 1, f+) = L(2m+ 1, f−)

which implies∞∑n=1

f+(n)− f−(n)

n2m+1= 0

for all m ∈ N.It follows that f+ = f− which shows that

L(1− s, f) =2Γ(s)√

(s)

( q

2π

)scos(πs

2

).

�

Using this fact and the fact G ∈ S#0 is a Dirichlet polynomial where

G(s) =∑n|Q2

g(n)

ns

with

g(n) = wn

Qg

(Q2

n

)and Q2 ∈ N as proved by Kaczowski and Perelli in their 1999 paper“On the Structure of the Selberg Class” we may prove another theorem.

Theorem 7. G ∈ S#0 if and only if there exists an f : N→ C where

f+ = wf

with w ∈ C, |w| = 1 and G(s) = L(s,f)ζ(s)

defines an entire function.

38

Proof. If G(s) = L(s,f)ζ(s)

is an entire function we can use the lemma to

show that if f satisfies f+ = wf with w ∈ C and |w| = 1 then

L(1− s, f) =2w√q

( q

2π

)sΓ(s) cos

(πs2

)L(s, f).

Using this and the fact that L(s, f) ∈ S#1 we get the functional equation

qs2G(s) = wq

(1−s)2 G(1− s).

As G(s) is the entire quotient of two series in S#1 it therefore lies in

S#0 .

Now to prove the opposite direction.

Assume G ∈ S0. There exists a Q2 ∈ N such that

G(s) =∞∑n=1

g(n)

ns

with g(n) = 0 if nQ2.

Now, if we define

f =∑d|n

g(d)

then

f(n+Q2) =∑

d|n+Q2

d|Q2

g(d) =∑d|nd|Q2

g(d) = f(n)

which shows f is Q2-periodic.

Now we must show that f+ = wf with w ∈ C, |w| = 1 and G(s) =L(s,f)ζ(s)

defines an entire function.

We know

L(1− s, f) =2√q

( q

2π

)sΓ(s) cos

(πs2

)L(s, f+)

=2w√q

( q

2π

)sΓ(s) cos

(πs2

)L(s, f)

which implies

L(s, f+) = wL(s, f).

This proves f+ = wf with w ∈ C and |w| = 1 and also proves thetheorem. �

39

5.8. Functions in the Selberg Class. The four functions we studiedin previous chapters can be shown to be in the Selberg Class.

First we will show the Riemann-zeta function is in the Selberg class.

Proof. (i) For the Riemann-zeta function we can clearly see that a1 = 1and an = 0 for all n ≤ 2. This is equivalent to showing an � nε. Fromsection 1.1 we know ζ(s) is absolutely convergent for Re(s) > 1 whichmeans property (i) holds.

(ii) We know ζ(s) is analytic except at s = 1. When s = 1 we have(s − 1)m = 0 which implies ζ(s) is analytic over the whole complexplane and therefore (ii) holds.

(iii) As shown in section 1.1 the functional equation is

ζ(s) = 2sπs−1Γ(1− s)sin(πs

2).

Therefore (iii) holds.

(iv) As shown in section 1.1 the Euler product is

ζ(s) =∏p

1

1− p−s.

Therefore (iv) holds and ζ(s) ∈ S.�

Now we will proove the Dirichlet L-functions are in the Selberg class.

Proof. (i) As stated in section 2.1 L(s, χ) is absolutely convergent forRe(s) > 1. Because an = χ(s) either equals 0 or 1 this means an ≤1� nepsilon. Also, a1 = 1 as g.c.d(m, 1) = 1. Therefore (i) holds.

(ii) L(S, χ) is analytic for all non-trivial χ. Therefore, (s−1)mL(s, χ)is analytic for m = 1 and (ii) holds.

(iii) As shown in section 2.4 the functional equation of L(s, χ) is

L(s, chi) = εL(1− s, χ∗).Therefore (iii) holds.

(iv) The Euler product of a Dirichlet L-function can be written as

L(s, χ) =∏p

(1− χpp−s)−1.

Therefore (iv) holds and L(s, χ) ∈ S. �

Now, as both the modular forms and elliptic curves can all haveconnected L-functions this shows they are also part of the Selberg class.

40

References

[1] Edwards, H.M., The Riemann-zeta function, 1974.[2] Rose, H.E., A Course in Number Theory, 1995.[3] Apostol, T, M., Introduction to Analytic Number Theory, 1976.[4] Apostol, T, M., Modular Functions and Dirichlet Series in Number Theory,

1990.[5] Hughes, C. and Beinek, J., Great Moments of the Riemann-zeta Function, 2008.[6] Koblitz, N., Introduction to elliptic curves and modular forms, 1984.[7] Selberg, A., Old and new conjectures and results about a class of Dirichlet

series, 1992.[8] Soundararajan, K., Degree 1 elements of the Selberg class, 2005.[9] Conrey, J, B. and Ghosh, A., On the Selberg Class of Dirichlet Series: Small

Degrees, 1993.[10] Kaczorowski, J. and Perelli, A., On the Structure of the Selberg Class, I : 0 ≤

D ≤ 1, 1999.[11] Kowalski, E. Michel, P. and VanderKam J., Rankin-Selberg L-functions in the

Level Aspect,[12] Steuding, J., A Note on the Selberg class (degrees 0 and 1), 2003.