What are the zeros of the Riemann zeta function?

The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6.. These are called its trivial zeros.

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 Riemann Hypothesis really solved?

While the distribution does not follow any regular pattern, Riemann believed that the frequency of prime numbers is closely related to an equation called the Riemann Zeta function. On the website of Clay Mathematics Institute, the final word on Riemann Hypothesis is: “The problem is unsolved”.

What happens if Riemann Hypothesis is true?

If the Riemann hypothesis is true, it won’t produce a prime number spectrometer. But the proof should give us more understanding of how the primes work, and therefore the proof might be translated into something that might produce this prime spectrometer.

Is Zeta a number?

Numeral. Zeta has the numerical value 7 rather than 6 because the letter digamma (also called ‘stigma’ as a Greek numeral) was originally in the sixth position in the alphabet.

Is the Riemann hypothesis solved?

What are trivial zeros?

The trivial zeros are the negative even integers –2, –4, –6, –8,…, so-called because it is relatively easy to demonstrate that ζ(2n) = 0 for natural numbers n. The nontrivial zeros are much more mysterious.

What is the hardest math question ever?

But those itching for their Good Will Hunting moment, the Guinness Book of Records puts Goldbach’s Conjecture as the current longest-standing maths problem, which has been around for 257 years. It states that every even number is the sum of two prime numbers: for example, 53 + 47 = 100. So far so simple.

How to calculate the zeroes of the Riemann zeta function?

Calculating the Zeroes of the Riemann-Zeta function. The Riemann zeta function $\\zeta(s)$ is defined for all complex numbers $s \ eq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6.)$. These are called the trivial zeros.

Is the Riemann zeta function holomorphic or meromorphic?

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. ζ ( 2 n ) = ( − 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) ! {\\displaystyle \\zeta (2n)= {\\frac { (-1)^ {n+1}B_ {2n} (2\\pi )^ {2n}} {2 (2n)!}}} where B2n is the 2n th Bernoulli number .

What are the values of the zeta function?

The values of the zeta function at non-negative even integers have the generating function : {\\displaystyle \\zeta (1)=1+ {\\frac {1} {2}}+ {\\frac {1} {3}}+\\cdots =\\infty \\!} ζ ( 3 ) = 1 + 1 2 3 + 1 3 3 + ⋯ = 1

What does the Riemann hypothesis say about non trivial zeros?

The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any non-trivial zero of the Riemann zeta function is 1 2. What does it mean to say that ζ(s) has a trivial zero and a non-trivial zero. I know that ζ(s) = ∞ ∑ n = 1 1 ns what wikipedia claims it that ζ( − 2) = ∑∞n = 1n2 = 0 which looks absurd.