Jensen polynomials for the Riemann zeta function and other sequences

Area of Science:

• Number Theory
• Mathematical Physics

Background:

• The Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function, a property proven for specific degrees.
• Proving hyperbolicity for all Jensen polynomials is crucial for understanding the Riemann zeta function and related mathematical conjectures.

Purpose of the Study:

• To establish an asymptotic formula for central derivatives of Jensen polynomials for the Riemann zeta function.
• To prove the hyperbolicity of Jensen polynomials for the Riemann zeta function for all degrees.
• To connect these findings to predictions from random matrix theory and resolve a conjecture on partition functions.

Main Methods:

• Deriving an asymptotic formula for central derivatives of Jensen polynomials, accurate to all orders.
• Modeling Jensen polynomials using Hermite polynomials within a general theorem.
• Applying the general theorem to the specific case of the Riemann zeta function.

Main Results:

• An asymptotic formula for central derivatives [Formula: see text] of Jensen polynomials for the Riemann zeta function is obtained.
• The hyperbolicity of all but finitely many Jensen polynomials of each degree is proven.
• Hyperbolicity is established for all Jensen polynomials [Formula: see text].

Conclusions:

• The study proves the hyperbolicity of Jensen polynomials for the Riemann zeta function, supporting the Riemann hypothesis.
• Results align with and confirm predictions from the Gaussian unitary ensemble random matrix model.
• A conjecture by Chen, Jia, and Wang concerning the partition function is also proven using the general theorem.