Sage-Tips

What does the Riemann zeta function tell us?

Riemann zeta function, function useful in number theory for investigating properties of prime numbers. Written as ζ(x), it was originally defined as the infinite series ζ(x) = 1 + 2−x + 3−x + 4−x + ⋯. For values of x larger than 1, the series converges to a finite number as successive terms are added.

Is the Riemann zeta function symmetrical?

As far as I learned from the literature, the non-trivial zeros of the zeta function are symmetric about the critical line Re(s) = 1/2, because xi(s) = xi(1-s). Instead the zeros are symmetric about Re(s) = 1/2 AND Im(s) = 0.

What does the Riemann zeta function have to do with primes?

The expression states that the sum of the zeta function is equal to the product of the reciprocal of one minus the reciprocal of primes to the power s. This astonishing connection laid the foundation for modern prime number theory, which from this point on used the zeta function ζ(s) as a way of studying primes.

What is Zeta in math?

zeta function, in number theory, an infinite series given by. where z and w are complex numbers and the real part of z is greater than zero.

How is Zeta calculated?

Integral Representation The zeta function can be represented as Γ ( s ) ζ ( s ) = ∫ 0 ∞ x s − 1 e x − 1 d x . \Gamma \left( s \right) \zeta \left( s \right) =\int _{ 0 }^{ \infty }{ \frac { { x }^{ s-1 } }{ e^x-1} dx }. Γ(s)ζ(s)=∫0∞​ex−1xs−1​dx.

What are trivial zeros?

The trivial zeros are simply the negative even integers. The nontrivial zeros are known to all lie in the critical strip 0 < Re[s] < 1, and always come in complex conjugate pairs. All known nontrivial zeros lie on the critical line Re[s] = 1/2. The Riemann Hypothesis states that they all lie on this line.

Are the nontrivial zeros of the Riemann zeta function statistical?

There exist important conjectures which relate the statistical behaviour of the nontrivial zeros of the Riemann zeta function to the statistical behaviour of the eigenvalues of large random matrices.

What is the distribution of Riemann zeros in the critical strip?

Asymptotically, the locations of the nontrivial zeros of the Riemann zeta function distribute in the critical strip according to a fairly simple logarithmic law, much like the primes do along the real line. Whereas the primes tend to become increasingly spaced out at a logarithmic rate, the zeta zeros tend to become more dense.

Is the Riemann zeta function a meromorphic function?

For s = 1, the series is the harmonic series which diverges to +∞, and lim s → 1 ( s − 1 ) ζ ( s ) = 1. {\\displaystyle \\lim _ {s o 1} (s-1)\\zeta (s)=1.} Thus the Riemann zeta function is a meromorphic function on the whole complex s -plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1 .

Are there any better estimates of zeta zeros for singularities?

There are no doubt better estimates but this is a rule-of-thumb. In the spirit of Raymond Manzoni’s answer in combination with the Franca-LeClair asymptotic of the zeta zeros here, one can arrive at the following almost identical zeta zero counting function: where only the singularities are different.