Mathematical challenges of heat equations in the twenty-first century

In the invited review paper we report the unsolved problems for the heat equations in theory of the Riemann zeta function. The Fourier cosine integral is derived from the Riemann xi function associated with the Riemann zeta function. The tempered xi function is one of the entire functions in analytic number theory. The Fourier sine integral is derived from the tempered xi function. The Fourier cosine and sine integrals can be considered as the initial value conditions of the heat equation. The deformed Fourier cosine and sine integrals related to the exponential function can be considered as the solutions of the heat equation. There exists a great many of mathematics conjectures related to the heat equations containing Riemann hypothesis, Jensen’s conjecture, and boldly asserts. They are still challenges for heat equations in the twenty-first century.