The Riemann Hypothesis for Siegel Modular Forms

Lauren Grainer (Bucknell University, Department of Mathematics) and Nathan Ryan (Bucknell University, Department of Mathematics) - Advisor

A Siegel modular form F(x,y,z) of degree 2 is a modular form in three variables.

We define the L-function:

L(F,s) := Π 1/(1-λpp-s+p-2s(λ2p-λp2-p2k-4)-λpp2k-s - p4k-6-p-4s )

where λp and λp2

are two real numbers easily computed from F.

As in the other two cases, we define a related function

Λ(F,s)=4(2п)-2s-k+1 Γ(s+1/2) Γ(s+k-3/2)

There are two kinds of Siegel modular forms: those that are lifts of modular forms and those that are not. The L-function of a lift F is imprimitive; i.e., L(F,s) can be written as the product of other L-functions. In particular, when F is a lift, it has zeroes where its factors do.

A graph of |Λ (Δ,½ + it)| for 0≤ t ≤ 30

Acknowledgments

•The research described above was supported by a 2008 Kalman Symposium Research Grant

A graph of |Λ (F,½ + it)| for 0≤ t ≤ 30 (red one) and a graph of |Λ (F,0+ it)| (blue one) when F is a lift

A graph of |Λ (F,½ + it)| for 0≤ t ≤ 30 when F is not a lift

A modular form is a function f(z) from the complex upper half plane to the complex plane.

The most famous example might be the following:

Δ(z):= Π q(1-qn)-24

= q-24q2+252q3+… = ∑n≥1 T(n)qn,

where q=e2пiz.

We define an L-function related to Δ via

L(Δ,s) := ∑n≥1 (Τ(n)/n11/2)/ns = Π (1-(T(p)/p11/2)p-s + p1-2s)-1

As in the original problem, defined a related function

Λ(Δ,s)=(2п)-s Γ (s + 11/2) L(Δ,s).

The Riemann Hypothesis says that all the non-trivial zeros are of the form ½ +it.

Abstract and Background Information

Statement of the Original Problem

Generalization II

Let

ζ(s) = ∑n≥1 1/(ns) = 1+ 1/(2s)+1/(3s)+1/(4s)+1/(5s)+...

By the Fundamental Theorem of Arithmetic:

ζ(s) = Π (1-p-s)-1 …

We define a related function

Λ(s) = QsΠΓ(kjS+λj) ζ(s)

The research focused on the Riemann hypothesis which is one of the most famous unsolved mathematical problems in history. It states that all the real parts of the zeros of normalized, primitive L-functions, including those related to modular forms, lie on the critical line x= ½. In this project we verified the Riemann Hypothesis for Classical Modular forms and Siegel Modular forms. Moreover we verified that imprimitive L-functions attached to lifted Siegel Modular forms violate the Riemann Hypothesis.

Bibliography

•Breulmann, Stefan and Michael Kuss.  On a conjecture of Duke-Imamoglu, Proceedings AMS, 2000.•Conrey, J. Brian.  The Riemann Hypothesis, Notices of the AMS, 2003.•Rubinstein, Michael.  Lcalc: An L-function calculator.•Ryan, Nathan C and Thomas R. Shemanske.  Inverting the Satake Isomorphism for Spn with an Application to Hecke Operators, Ramanujan J, 2009.•Stein, William.  Sage: Open Source Software for Mathematics.

The Riemann Hypothesis says that all non-trivial zeros of Λ(s) all are in the form of s= ½ +it

Generalization II