The Riemann zeta function (also known as the Euler–Riemann zeta function), notated as , is a function used in complex analysis and number theory. It is defined as the analytic continuation of the series
which converges for all s such that .
The Riemann hypothesis states that iff s is a negative even integer or the imaginary part of s is 1/2.
Representation as an integral
The Riemann zeta function can be expressed as an improper integral. Consider the improper integral,
One may multiply through to obtain,
From which point, considering,
One may rewrite the integrand as,
Which can be written as, by changing the order of the operators
By using a substitution of, say, , one can write this as
Notice that is independent of x, and is independent of n, meaning that our integral can be written as,
Using the definitions of both the zeta and the gamma function, we finally have,
The zeta function has no zeros where s is greater than or equal to one. When s is less than or equal to zero, the function has zeros on even integers known as trivial zeros. The remaining zeros are between zero and one; this is known as the critical strip. The Riemann hypothesis is that all non-trivial zeros lie on the line as ranges over all real numbers. Whether or not this is true is known as the Riemann Hypothesis.
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