The motivation for studying -functions and their zeros is to
gain information about the prime numbers. Thus, it is a fundamental
problem to describe the precise connection between zeros of
-functions and various properties of the prime numbers.
At present, the connection is fully understood in only a few cases,
such as the equivalence between the error term in the Prime Number Theorem
and the real part of the zeros of . Namely,
is equivalent to
for
.
The connection between -correlations of zeros of the
Riemann
-function
and the distribution of the prime numbers appears to be close to
being understood. See the article on
the distribution of primes for a discussion.
Most of the questions about the prime numbers of a special
form (twin primes, etc) haven't been shown to be related to
standard conjectures about the distribution of zeros of -functions.
See the articles on primes of a special form and
gaps between primes for a further discussion.
The Riemann -function can be expressed as a product
over the primes and also as a sum over its zeros. Thus, everything
about the prime numbers can be determined from the zeros of the
-function. However, not all information about the primes
can be extracted from the zeros in a simple way.
It is possible that some questions about the primes (eg, the variation
of the distribution of primes in short intervals) are naturally
related to the distribution of zeros of the Riemann
-function,
while other questions (twin primes?) may more naturally be related to
the distribution of zeros of Dirichlet
-functions. Also, some of
those questions may require information beyond the
GUE Hypothesis. It would be good to have some general
principles describing which properties of the primes are most
naturally related
to which properties of particular
-functions.
In this section we write for a prime and
for the next
larger prime. We also use the standard notation for the von Mangoldt function
if
and
otherwise,
and the Chebyshev function
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