Skewes' number

In number theory, Skewes's number is any of several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x {displaystyle x} for which In number theory, Skewes's number is any of several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x {displaystyle x} for which where π is the prime-counting function and li is the logarithmic integral function. There is a crossing near e 727.95133 < 1.397 × 10 316 . {displaystyle e^{727.95133}<1.397 imes 10^{316}.} It is not known whether it is the smallest. John Edensor Littlewood, who was Skewes's research supervisor, had proved in Littlewood (1914) that there is such a number (and so, a first such number); and indeed found that the sign of the difference π ( x ) − li ⁡ ( x ) {displaystyle pi (x)-operatorname {li} (x)} changes infinitely often. All numerical evidence then available seemed to suggest that π ( x ) {displaystyle pi (x)} was always less than li ⁡ ( x ) {displaystyle operatorname {li} (x)} . Littlewood's proof did not, however, exhibit a concrete such number x {displaystyle x} . Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x {displaystyle x} violating π ( x ) < li ⁡ ( x ) , {displaystyle pi (x)<operatorname {li} (x),} below In Skewes (1955), without assuming the Riemann hypothesis, Skewes proved that there must exist a value of x {displaystyle x} 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 Lehman (1966), who showed that somewhere between 1.53 × 10 1165 {displaystyle 1.53 imes 10^{1165}} and 1.65 × 10 1165 {displaystyle 1.65 imes 10^{1165}} there are more than 10 500 {displaystyle 10^{500}} consecutive integers x {displaystyle x} with π ( x ) > li ⁡ ( x ) {displaystyle pi (x)>operatorname {li} (x)} .Without assuming the Riemann hypothesis, H. J. J. te Riele (1987) proved an upper bound of 7 × 10 370 {displaystyle 7 imes 10^{370}} . A better estimate was 1.39822 × 10 316 {displaystyle 1.39822 imes 10^{316}} discovered by Bays & Hudson (2000), who showed there are at least 10 153 {displaystyle 10^{153}} consecutive integers somewhere near this value where π ( x ) > li ⁡ ( x ) , {displaystyle pi (x)>operatorname {li} (x),} and suggested that there are probably at least 10 311 {displaystyle 10^{311}} . Bays and Hudson found a few much smaller values of x {displaystyle x} where π ( x ) {displaystyle pi (x)} gets close to li ⁡ ( x ) {displaystyle operatorname {li} (x)} ; 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. Chao & Plymen (2010) gave a small improvement and correction to the result of Bays and Hudson. Saouter & Demichel (2010) found a smaller interval for a crossing, which was slightly improved by Zegowitz (2010). The same source shows that there exists a number x {displaystyle x} violating π ( x ) < li ⁡ ( x ) , {displaystyle pi (x)<operatorname {li} (x),} below e 727.9513468 < 1.39718 × 10 316 {displaystyle e^{727.9513468}<1.39718 imes 10^{316}} . This can be reduced to e 727.9513386 < 1.39717 × 10 316 {displaystyle e^{727.9513386}<1.39717 imes 10^{316}} , assuming the Riemann hypothesis. Stoll & Demichel (2011) gave 1.39716 × 10 316 {displaystyle 1.39716 imes 10^{316}} . Rigorously, Rosser & Schoenfeld (1962) proved that there are no crossover points below x = 10 8 {displaystyle x=10^{8}} , improved by Brent (1975) to 8 × 10 10 {displaystyle 8 imes 10^{10}} , by Kotnik (2008) to 10 14 {displaystyle 10^{14}} , by Platt & Trudgian (2014) to 1.39 × 10 17 {displaystyle 1.39 imes 10^{17}} , and by Büthe (2015) to 10 19 {displaystyle 10^{19}} . There is no explicit value x {displaystyle x} known for certain to have the property π ( x ) > li ⁡ ( x ) , {displaystyle pi (x)>operatorname {li} (x),} though computer calculations suggest some explicit numbers that are quite likely to satisfy this.