Hadamard's proof depends on the fact that the quantity one plus cosine theta squared is greater than or equal to zero, and follows from the fact that the Riemann zeta function has no roots of the form one plus i times t, where i is the square root of minus one.
Ingham proved it using Ramanujan's identity, and it is equivalent to the statement that the limit as x goes to infinity of theta of x over x or psi of x over x is one, where theta and psi are the Chebyshev functions. Hadamard's proof depends on the fact that the quantity one plus cosine theta squared is greater than or equal to zero, and follows from the fact that the Riemann zeta function has no roots of the form one plus i times t, where i is the square root of minus one. FTP, name this theorem which gives x over natural log x as an approximation to the number of non-composite integers less than or equal to x, which takes its name from those non-compposite integers.
Ingham proved it using Ramanujan's identity, and it is equivalent to the statement that the limit as x goes to infinity of theta of x over x or psi of x over x is one, where theta and psi are the Chebyshev functions.
prime number theorem