Riemann zeta function

The Riemann zeta function is initially defined by the Dirichlet seriesUnique prime factorization gives its Euler productExpanding each factor as a geometric series chooses an exponent for each prime number, so each positive integer appears exactly once. Absolute convergence justifies this rearrangement. This identity makes the Riemann zeta function a bridge between the distribution of prime numbers and complex analysis.

The function extends meromorphically to the complex plane, with its only pole at $s=1$, simple with residue $1$. For example, continuation gives $\zeta (0)=-1/2$ even though the defining positive-term series diverges there. The functional equation of the Riemann zeta function relates values at $s$ and $1-s$ through the Riemann xi function.^[ref]

For $\mathrm{Re}s>1$, its negative logarithmic derivative has coefficients given by the Von Mangoldt function. Perron's formula transfers analytic information about its zeros into estimates for prime-counting functions. The Riemann hypothesis specifies the location of its Nontrivial zeros of the Riemann zeta function.

The Riemann zeta function, denoted as \(\zeta(s)\), is a complex function defined for complex numbers \(s = \sigma + it\), where \(\sigma\) and \(t\) are real numbers.