Riemann hypothesis for period polynomials of modular forms

Area of Science:

• Number Theory
• Automorphic Forms
• Analytic Number Theory

Background:

• The period polynomial r(f)(z) is defined for newforms f of even weight k≥4.
• It serves as a generating function for critical values of the L-function L(f,s).
• A functional equation relates r(f)(z) to r(f)(-1/Nz).

Purpose of the Study:

• To prove the Riemann hypothesis for the period polynomials of newforms.
• To investigate the distribution of the zeros of these polynomials.

Main Methods:

• Utilizing the functional equation of the period polynomial.
• Applying techniques from analytic number theory to analyze the zeros.

Main Results:

• The Riemann hypothesis is proven for the period polynomial r(f)(z), showing its zeros lie on the circle |z|=1/√N.
• Equidistribution of these zeros is established when the weight k or the level N is large.

Conclusions:

• The study provides a significant result regarding the distribution of zeros of period polynomials.
• These findings contribute to the understanding of L-functions and their properties in number theory.