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  2. Riemann hypothesis - Wikipedia

    en.wikipedia.org/wiki/Riemann_hypothesis

    Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.

  3. Generalized Riemann hypothesis - Wikipedia

    en.wikipedia.org/wiki/Generalized_Riemann_hypothesis

    The Riemann hypothesis is one of the most important conjectures in mathematics.It is a statement about the zeros of the Riemann zeta function.Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function.

  4. Explicit formulae for L-functions - Wikipedia

    en.wikipedia.org/wiki/Explicit_formulae_for_L...

    The other terms also correspond to zeros: The dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions.

  5. Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Riemann_zeta_function

    The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line. [9] This has since been improved to 41.7%. [10]

  6. On the Number of Primes Less Than a Given Magnitude

    en.wikipedia.org/wiki/On_the_Number_of_Primes...

    The entire function ξ(s), related to the zeta function through the gamma function (or the Π function, in Riemann's usage) The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power p n. (Riemann calls this function f(x).) Among the proofs and sketches of proofs:

  7. 10 Hard Math Problems That Even the Smartest People in the ...

    www.aol.com/10-hard-math-problems-even-150000090...

    Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is, “Every nontrivial zero of the Riemann zeta function has real part 1/2.”

  8. Hilbert's eighth problem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_eighth_problem

    It asks for more work on the distribution of primes and generalizations of Riemann hypothesis to other rings where prime ideals take the place of primes. Absolute value of the ζ-function. Hilbert's eighth problem includes the Riemann hypothesis, which states that this function can only have non-trivial zeroes along the line x = 1/2 [2].

  9. Particular values of the Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Particular_values_of_the...

    The Riemann hypothesis states that the real part of every nontrivial zero must be ⁠ 1 / 2 ⁠. In other words, all known nontrivial zeros of the Riemann zeta are of the form z = ⁠ 1 / 2 ⁠ + yi where y is a real number. The following table contains the decimal expansion of Im(z) for the first few nontrivial zeros: