camilus said:positive. The RZF has a pole at s=1 because at s=1 it is equal to the harmonic series sum 1+1/2+1/3+1/4+1/5+...= infinity
The Riemann zeta function, denoted by ζ(s), is a mathematical function that was first defined and studied by Bernhard Riemann in the 19th century. It is defined for all complex numbers s with a real part greater than 1, and is a fundamental tool in number theory and theoretical physics.
The Riemann zeta function is significant because it has connections to many important areas of mathematics, including number theory, complex analysis, and harmonic analysis. It also plays a crucial role in the study of prime numbers, as it relates to the distribution of prime numbers along the number line.
The Riemann hypothesis is a famous unsolved problem in mathematics, first proposed by Bernhard Riemann in 1859. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line, which is the line where the real part of s is equal to 1/2. The Riemann hypothesis has many important implications in number theory, and its proof or disproof remains one of the most important open problems in mathematics.
The Riemann zeta function pole question is a specific aspect of the Riemann hypothesis, which asks whether or not the Riemann zeta function has any poles (singularities) other than the expected ones on the critical line. This question is closely related to the overall validity of the Riemann hypothesis and has been the subject of much research and speculation.
The Riemann zeta function pole question is important because its answer is crucial to understanding the behavior of the Riemann zeta function and ultimately to proving or disproving the Riemann hypothesis. It also has implications for other areas of mathematics, such as the distribution of prime numbers and the study of other types of zeta functions. Resolving this question could lead to a deeper understanding of the connections between different branches of mathematics.