The motivation for studying the zeros of the zeta-function
is the precise relationship between the zeros of and
the errot term in the prime number theorem.
Define the Von Mangoldt function by

where is a prime number. Then

if and only if does not vanish for . There is the logarithmic integral

In particular, the best possible error term in the prime number theorem is , which is equivalent to the Riemann Hypothesis.

Similarly, the Generalized Riemann Hypothesis is equivalent to the best possible error term for the counting function of primes in arithmetic progressions.

Riemann Hypotheses for other -functions have not yet been shown to be strongly connected to the distribution of the prime numbers.

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for The Riemann Hypothesis.