lpetrich
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Famed mathematician claims proof of 160-year-old Riemann hypothesis | New Scientist
The mathematician is Michael Atiyah. Born in 1929, he is now retired, but he continues to be active. He has won some great awards for his mathematical work.
The
Riemann hypothesis is a conjecture about the possible values of the zeros of the Riemann zeta function:
\( \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} =\frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}}{e^x - 1} dx\)
The product is over all prime numbers p. Though it may seem that the Riemann zeta function is only defined for Re(s) > 1 (Re = real part), one can do a common mathematical trick called "analytic continuation" to find its value for other values of s. When one does so, one finds that this function is zero for all even negative integers.
The conjecture states that all the other zeros of the Riemann zeta function have real part 1/2. It is related in a certain way to the distribution of prime numbers, and it has long been an unsolved problem. From the article, "New Scientist contacted a number of mathematicians to comment on the claimed proof, but all of them declined. Atiyah has produced a number of papers in recent years making remarkable claims which have so far failed to convince his peers." So other mathematicians will have to take a look at his claimed proof.
The Riemann hypothesis is one of the seven Clay Millennium Prize problems in mathematics, and if Michael Atiyah's proof is valid, he could win $1 million in prize money. He will join Grigori Perelman, who has solved the Poincaré conjecture.
The mathematician is Michael Atiyah. Born in 1929, he is now retired, but he continues to be active. He has won some great awards for his mathematical work.
The
Riemann hypothesis is a conjecture about the possible values of the zeros of the Riemann zeta function:\( \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} =\frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}}{e^x - 1} dx\)
The product is over all prime numbers p. Though it may seem that the Riemann zeta function is only defined for Re(s) > 1 (Re = real part), one can do a common mathematical trick called "analytic continuation" to find its value for other values of s. When one does so, one finds that this function is zero for all even negative integers.
The conjecture states that all the other zeros of the Riemann zeta function have real part 1/2. It is related in a certain way to the distribution of prime numbers, and it has long been an unsolved problem. From the article, "New Scientist contacted a number of mathematicians to comment on the claimed proof, but all of them declined. Atiyah has produced a number of papers in recent years making remarkable claims which have so far failed to convince his peers." So other mathematicians will have to take a look at his claimed proof.
The Riemann hypothesis is one of the seven Clay Millennium Prize problems in mathematics, and if Michael Atiyah's proof is valid, he could win $1 million in prize money. He will join Grigori Perelman, who has solved the Poincaré conjecture.
