where are the zeros of zeta of s?
The zeros of the Riemann zeta function, denoted by ζ(s), are an interesting and important topic in number theory and complex analysis. The Riemann zeta function is defined for complex numbers s with a real part greater than 1, i.e., Re(s) > 1.
To find the zeros of the Riemann zeta function, we need to solve the equation ζ(s) = 0. However, this equation is not easy to solve analytically for general values of s.
One of the most famous unsolved problems in mathematics is the Riemann Hypothesis, which states that all nontrivial zeros of the Riemann zeta function lie on the "critical line" with real part 1/2. This critical line is the vertical line in the complex plane where the real part of s equals 1/2.
In other words, if we assume the Riemann Hypothesis is true, then all the nontrivial zeros of the Riemann zeta function lie on the line Re(s) = 1/2.
However, there are some known zeros of the Riemann zeta function outside the critical line, which are called "trivial zeros." The trivial zeros occur when s is a negative even integer, where ζ(s) is undefined and equal to zero.
To summarize, the nontrivial zeros of the Riemann zeta function are believed to lie on the critical line with real part 1/2, according to the Riemann Hypothesis. The trivial zeros occur when s is a negative even integer. However, determining the exact locations of these zeros is still an open question in mathematics.