Skewes's number

In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number for which where π is the prime-counting function and li is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing between and near It is not known whether it is the smallest crossing. J.E. Littlewood, who was Skewes's research supervisor, had proved in that there is such a number (and so, a first such number); and indeed found that the sign of the difference changes infinitely many times. All numerical evidence then available seemed to suggest that was always less than Littlewood's proof did not, however, exhibit a concrete such number . proved that, assuming that the Riemann hypothesis is true, there exists a number violating below In , without assuming the Riemann hypothesis, Skewes proved that there must exist a value of below Skewes's task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to Georg Kreisel, this was at the time not considered obvious even in principle. These upper bounds have since been reduced considerably by using large-scale computer calculations of zeros of the Riemann zeta function. The first estimate for the actual value of a crossover point was given by , who showed that somewhere between and there are more than consecutive integers with . Without assuming the Riemann hypothesis, proved an upper bound of . A better estimate was discovered by , who showed there are at least consecutive integers somewhere near this value where . Bays and Hudson found a few much smaller values of where gets close to ; the possibility that there are crossover points near these values does not seem to have been definitely ruled out yet, though computer calculations suggest they are unlikely to exist. gave a small improvement and correction to the result of Bays and Hudson. found a smaller interval for a crossing, which was slightly improved by .

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Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named.

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In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by pi(x) (unrelated to the number pi). Prime number theorem Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately where log is the natural logarithm, in the sense that This statement is the prime number theorem.

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In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value . The logarithmic integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral Here, ln denotes the natural logarithm.

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