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Since the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that . Brad Rodgers and Terence Tao discovered the equivalence is actually = by proving zero to be the lower bound of the constant. [16]
In 2018, with Brad Rodgers, Tao showed that the de Bruijn–Newman constant, the nonpositivity of which is equivalent to the Riemann hypothesis, is nonnegative. [32] In 2020, Tao proved Sendov's conjecture , concerning the locations of the roots and critical points of a complex polynomial, in the special case of polynomials with sufficiently ...
In September 2019, news broke regarding progress on this 82-year-old question, thanks to prolific mathematician Terence Tao. ... If the Riemann Hypothesis were solved tomorrow, it would unlock an ...
Here is a sketch of the proof referred to in one of Terence Tao's lectures. [15] ... One can even prove an analogue of the Riemann hypothesis, namely that
The constant is closely connected with Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the conjecture that . [1] Brad Rodgers and Terence Tao proved that , so the Riemann hypothesis is equivalent to =. [2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner. [3]
In 1859 Bernhard Riemann used complex analysis and a special meromorphic function now known as the Riemann zeta function to derive an analytic expression for the number of primes less than or equal to a real number x. Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture.
In the study of the Riemann hypothesis, a Lehmer pair is a pair of zeros of the Riemann zeta function that are unusually close to each other. [1] They are named after Derrick Henry Lehmer, who discovered the pair of zeros
Number theorists Paul Erdős and Terence Tao in 1985, when Erdős was 72 and Tao was 10 Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually ...