en>AnomieBOT: Dating maintenance tags: {{Cit}}

2014-08-15T21:22:09Z

Dating maintenance tags: {{Cit}}

← Older revision		Revision as of 23:22, 15 August 2014
Line 1:		Line 1:
	~~In [[number theory]], '''Skewes' number''' is any~~ of ~~several extremely large numbers used by the [[South Africa]]n mathematician [[Stanley Skewes]] as [[upper bound]]s for~~ the ~~smallest [[natural number]] ''x'' for which~~		Common history of the author is Gabrielle Lattimer. For years she's been working whereas a library assistant. For a while she's yet been in Massachusetts. As a woman what this lady really likes is mah jongg but she have not made a dime with it. She has become running and maintaining the best blog here: http://prometeu.net<br><br>Also visit my weblog ... [http://prometeu.net clash of clans hack tool download free]
	~~:<math>\pi(x) > \operatorname{li}(x),</math>~~
	~~where π is the [[prime-counting function]] and li~~ is ~~the [[logarithmic integral function]]~~. ~~These bounds have since been improved by others: there is a crossing near <math>e^{727.95133}</math>. It is not known whether it is the smallest.~~

	~~==Skewes' numbers==~~
	[[John Edensor Littlewood]], Skewes' teacher, proved (in {{harv\|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'') changes infinitely often. All numerical evidence then available seemed to suggest that π(''x'') is always less than li(''x''), though mathematicians familiar with Riemann's work on the [[Riemann zeta function]] would probably have realized that occasional exceptions were likely by the argument [[#Riemann's formula\|given below]] (and the claim sometimes made that Littlewood's result was a big surprise to experts seems doubtful). Littlewood's proof did not, however, exhibit a ~~concrete such number ''x''~~.

	~~{{harvtxt\|Skewes\|1933}} proved that, assuming that the [[Riemann hypothesis]] is true, there exists~~ a ~~number ''x'' violating π(''x'') < li(''x'') below~~
	~~:<math>e^{e^{e^{79}}}<10^{10^{10^{34}}}.</math>~~

	~~In {{harv\|Skewes\|1955}}, without assuming the Riemann hypothesis, Skewes managed to prove that there must exist a value of ''x'' below~~
	~~:<math>e^{e^{e^{e^{7.705}}}}<10^{10^{10^{963}}}.</math>~~

	~~Skewes' task was to make Littlewood~~'s existence proof effective: exhibiting some concrete upper bound for the first sign change. According to [[George Kreisel]], this was at the time not considered obvious even in principle. The approach called ''[[Proof mining\|unwinding]]'' in [[proof theory]] looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

	~~Although both Skewes' numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as [[Graham's number]].~~

	~~==More recent estimates==~~
	~~These (enormous) 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 {{harvtxt\|Lehman\|1966}}, who~~
	~~showed that somewhere between 1.53{{e\|1165}} and 1.65{{e\|1165}} there are more than 10<sup>500</sup> consecutive integers ''x'' with π(''x'') > li(''x'').~~
	Without assuming the Riemann hypothesis, {{harvs\|txt=yes\|authorlink=Herman te Riele\|first=H. J. J.\|last= te Riele \|year= 1987}} proved an upper bound of 7{{e\|370}}. A better estimation was 1.39822{{e\|316}} discovered by {{harvtxt\|Bays\|Hudson\|2000}}, who showed there are at least 10<sup>153</sup> consecutive integers somewhere near this value where π(''x'') > li(''x''), and suggested that there are probably at least 10<sup>311</sup>. {{harvtxt\|Chao\|Plymen\|2010}} gave a small improvement and correction to the result of Bays and Hudson. Bays and Hudson found a few much smaller values of ''x'' where π(x) gets close to 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. {{harv\|Saouter\|Demichel\|2010}} find a smaller interval for a crossing, which was slightly improved by {{harv\|Zegowitz\|2010}}. The same source shows that there exists a number ''x'' violating π(''x'') < li(''x'') below <math>e^{727.951346801}</math>. The exponent could be reduced to 727.951338611, assuming Riemann hypothesis.

	~~Rigorously, {{harvtxt\|Rosser\|Schoenfeld\|1962}} proved that there are no crossover points below ''x'' = 10<sup>8</sup>, and~~ this ~~lower bound was subsequently improved by {{harvtxt\|Brent\|1975}} to 8{{e\|10}}, and by {{harvtxt\|Kotnik\|2008}} to 10<sup>14</sup>.~~

	~~There~~ is ~~no explicit value ''x'' known for certain to~~ have ~~the property π(x) > li(x), though computer calculations suggest some explicit numbers that are quite likely to satisfy this.~~

	{{harvtxt\|Wintner\|1941}} showed that the [[natural density\|proportion of integers]] for which π(''x'')-li(''x'') is positive, and {{harvtxt\|Rubinstein\|Sarnak\|1994}} showed that this proportion is about .00000026, which is surprisingly large given how far one has to go to find the first example.

	~~==Riemann's formula==~~
	~~Riemann gave an [[Explicit formulae (L-function)\|explicit formula]] for π(x), whose leading terms are (ignoring some subtle convergence questions)~~

	~~:<math>\pi(x) = \operatorname{li}(x) - \frac{\operatorname{li}(\sqrt{x})}{2} - \sum_\rho \operatorname{li}(x^\rho) + \text{smaller terms} </math>~~
	where the sum is over zeros ρ of the Riemann zeta function. The largest error term in the approximation π(''x'') = li(x) (if the [[Riemann hypothesis]] is true) is li({{radic\|''x''}})/2, showing that li(''x'') is usually larger than π(x). The other terms above are somewhat smaller, and moreover tend to have different complex arguments so mostly cancel out. Occasionally however, many of the larger ones might happen to have roughly the same complex argument, in which case they will reinforce each other instead of cancelling and will overwhelm the term li({{radic\|''x''}})/2. The reason why the Skewes number is so large is that these smaller terms are quite a lot smaller than the leading error term, mainly because the first complex zero of the zeta function has quite a large imaginary part, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term. The chance of ''N'' random complex numbers having roughly the same argument is about 1 in 2<sup>''N''</sup>. This explains why π(''x'') is sometimes larger than li(''x''), and also why it is rare for this to happen. It also shows why finding places where this happens depends on large scale calculations of millions of high precision zeros of the Riemann zeta function. The argument above is not a ~~proof, as~~ it ~~assumes the zeros of the Riemann zeta function are random which is not true~~. ~~Roughly speaking, Littlewood's proof consists of [[Dirichlet's approximation theorem]] to show that sometimes many terms have about~~ the ~~same argument.~~

	In the event that the Riemann hypothesis is false, the argument is much simpler, essentially because the terms li(''x''<sup>ρ</sup>) for zeros violating the Riemann hypothesis (with real part greater than 1/2) are eventually larger than li(''x''<sup>1/2</sup>).

	The reason for the term <math>\mathrm{li}(x^{1/2})/2</math> is that, roughly speaking, <math>\mathrm{li}(x)</math> counts not primes, but powers of primes <math>p^n</math> weighted by <math>1/n</math>, and <math>\mathrm{li}(x^{1/2})/2</math> is a sort of correction term coming from squares of primes.

	~~==References==~~
	~~{{reflist}}~~
	~~{{refbegin}}~~
	*{{citation\|mr=1752093\|first=C.\|last= Bays \|first2=R. H.\|last2= Hudson \|url=http://~~www~~.ams.org/mcom/2000-69-231/S0025-5718-99-01104-7/S0025-5718-99-01104-7.pdf \|title=A new bound for the smallest ''x'' with π(''x'') > li(''x'') \|journal=[[Mathematics of Computation]]\|volume=69\|year=2000 \|issue= 231\|pages= 1285–1296}}
	*{{citation\|mr=0369287\|first=R. P.\|last= Brent \|title=Irregularities in the distribution of primes and twin primes \|journal=[[Mathematics of Computation]]\|volume=29\|year=1975 \|pages= 43–56\|doi=10.2307/2005460\|jstor=2005460\|issue=129}}
	*{{citation\|unused_data=year=2010\|doi=10.1142/S1793042110003125\|title=A new bound for the smallest ''x'' with {{math\|π(''x'') > li(''x'')}}\|first=Kuok Fai\|last= Chao\|first2= Roger\|last2= Plymen\|year=2010\|arxiv=math/0509312 \|journal= International Journal of Number Theory \|volume= 6\|issue=03\|pages= 681–690\|mr=2652902}}
	*{{citation\|first= T.\|last= Kotnik \|doi=10.1007/s10444-007-9039-2 \|title=The prime-counting function and its analytic approximations \|journal=Advances in Computational Mathematics\|volume=29\|issue= 1\|year=2008\|pages= 55–70}}
	*{{citation\|first= R. Sherman \|last=Lehman\|title= On the difference π(''x'') − li(''x'')\|journal= [[Acta Arithmetica]] \|volume=11 \|year=1966\|pages= 397–410 \|mr=0202686}}
	* {{citation\|first=J. E.\|last= Littlewood\|title=Sur la distribution des nombres premiers\|journal=[[Comptes Rendus]]\|volume= 158 \|year=1914\|pages= 1869–1872}}
	*{{citation\|first= S.\|last= Skewes\|title=On the difference π(''x'') − Li(''x'')\|journal=[[Journal of the London Mathematical Society]]\|volume=8\|year=1933\|pages= 277–283}}
	*{{citation\|mr=0067145\| first= S.\|last= Skewes\|title=On the difference π(''x'') − Li(''x'') (II)\|journal=[[Proceedings of the London Mathematical Society]]\|volume= 5 \|year=1955\|pages= 48–70}}
	*{{citation\|mr=0866118\|first= H. J. J. \|last=te Riele\|title=On the sign of the difference π(''x'') − Li(''x'')\|journal=[[Mathematics of Computation]]\|volume=48\|year=1987\|pages= 323–328 \|jstor=2007893 \|issue=177 }}
	*{{citation\|mr=0137689\|first= J. B.\|last= Rosser \|first2= L.\|last2= Schoenfeld \|title=Approximate formulas for some functions of prime numbers \|journal=Illinois Journal of Mathematics\|volume=6\|year=1962\|pages= 64–94}}
	*{{citation
	~~\| last1 = Saouter \| first1 = Yannick~~
	~~\| last2 = Demichel \| first2 = Patrick~~
	~~\| doi = 10.1090/S0025-5718-10-02351-3~~
	~~\| mr = 2684372~~
	~~\| issue = 272~~
	~~\| journal = [[Mathematics of Computation]]~~
	~~\| pages = 2395–2405~~
	~~\| title = A sharp region where {{math\|π(''x'') − li(''x'')}} is positive~~
	~~\| volume = 79~~
	~~\| year = 2010}}~~
	*{{citation\|last=Zegowitz\|first=Stefanie\|title=On the positive region of <~~math~~>~~\pi(x)-\operatorname{li}(x)~~<~~/math~~>~~\|pages=69 pp.\|year=2010\|url=http://eprints.ma.man.ac.uk/1547/}}~~
	*{{citation\|mr=1329368 \|author2-link=Peter Sarnak \|last=Rubinstein\|first= M.~~\|last2= Sarnak\|first2= P. \|title=Chebyshev's bias~~
	~~\|journal=[[Experimental Mathematics (journal)\|Experimental Mathematics]] \|volume=3 \|year=1994\|issue= 3\|pages= 173–197 \|url= http://projecteuclid~~.~~org/euclid~~.~~em/1048515870 }}~~
	*{{citation\|mr=0004255\|last= Wintner\|first= A. \|title=On the distribution function of the remainder term of the prime number theorem \|journal= [[American Journal of Mathematics]]\|volume= 63\|year=1941\|pages= 233–248\|issue=2\|doi=10.2307/2371519\|jstor=2371519\|publisher=The Johns Hopkins University Press }}
	~~{{refend}}~~

	~~==External links==~~
	*[http://~~web.archive.org/web/20060908033007/http://demichel~~.net~~/patrick/li_crossover_pi.pdf The prime counting function and related subjects] by Patrick Demichels (retrieved 2009-09-29)~~

	~~{{Large numbers}}~~

	~~[[Category:Large numbers]]~~
	~~[[Category:Number theory]]~~
	~~[[Category:Large integers]~~]

en>Monkbot: /* Early discovery and genetic equidistance */Fix CS1 deprecated date parameter errors

2014-01-26T20:48:46Z

Early discovery and genetic equidistance: Fix CS1 deprecated date parameter errors

Show changes

en>RedBot: r2.7.2) (Robot: Adding nl:Moleculaire klok

2012-08-09T19:58:40Z

r2.7.2) (Robot: Adding nl:Moleculaire klok

New page

You could have attempted plus failed when we have attempted to drop some fat. If you manage to shed a number of pounds you then have the complicated task of keeping the weight off. We are probably following the incorrect rules that is why the fat begins to grow again. Losing fat isn't too difficult plus really it is actually simple to do away with it and keep it off in the event you learn extremely well what you're undertaking. Below I will inform we techniques to drop weight effectively plus the method to keep it off. Don't provide [http://safedietplansforwomen.com/bmr-calculator bmr calculator] up if you blow your diet for the day. Tomorrow is a hot day for eating healthy foods. The worst thing we can do is beat oneself up with guilt after overeating or bingeing, plus you're many vulnerable whenever anxious or depressed. In daily life calories are burned for different reasons. Even at rest calories are burnt which is represented by basal metabolic rate (BMR). For imperative activities like breathing, blood circulation to be maintained, the body needs stamina. By eating, digesting and storing food calories are burned, that is termed because Dietary thermogenesis. Our bodies want 1,800 to 2,000 calories a day in purchase to function properly. Some individuals want less or even more, however, this might be the average. So don't try to drop a significant amount of calories at once. There is no should place the body by starvation. If you do, your metabolic rate really slows down, that makes the entire procedure harder. Then let's calculate the amount of calories a have to support your daily escapades. Simply multiply the calculated bmr by 1.2 when you don't exercise at all, 1.375 in the event you exercise lightly 1 to 3 instances per week, 1.55 should you exercise moderately 3 to 5 times per week, 1.725 in the event you exercise hard 6 to 7 occasions per week, or 1.9 if you have a physically demanding job and exercise every day or are training for a sports competition like a marathon. At any provided moment, 25 % of all guys plus 33 % of all ladies are on several sort of formal diet in the United States. More than 55 percent gain back all of their weight and more than what they started with.1 Unfortunately, many diets are a one-size-fits-all approach. With any diet book you choose off the bookstore shelf, or any aged diets passed down by the good aunt, there are the same diet for everyone. Some of those are completely unsound nutritionally while others will be backed by superior nutrition principles. Yet, even those with advantageous nutrition principles don't personalize their approach to fit every person's body makeup. These are typically regrettably a one-size-fits-all dieting approach. I hope you may be inside the regular range, nevertheless in the event you are overweight we can plan some fat loss system considering the BMR and present activity level and hopefully boost a wellness.

← Older revision		Revision as of 23:22, 15 August 2014
Line 1:		Line 1:
	~~In [[number theory]], '''Skewes' number''' is any~~ of ~~several extremely large numbers used by the [[South Africa]]n mathematician [[Stanley Skewes]] as [[upper bound]]s for~~ the ~~smallest [[natural number]] ''x'' for which~~		Common history of the author is Gabrielle Lattimer. For years she's been working whereas a library assistant. For a while she's yet been in Massachusetts. As a woman what this lady really likes is mah jongg but she have not made a dime with it. She has become running and maintaining the best blog here: http://prometeu.net<br><br>Also visit my weblog ... [http://prometeu.net clash of clans hack tool download free]
	~~:<math>\pi(x) > \operatorname{li}(x),</math>~~
	~~where π is the [[prime-counting function]] and li~~ is ~~the [[logarithmic integral function]]~~. ~~These bounds have since been improved by others: there is a crossing near <math>e^{727.95133}</math>. It is not known whether it is the smallest.~~

	~~==Skewes' numbers==~~
	[[John Edensor Littlewood]], Skewes' teacher, proved (in {{harv\|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'') changes infinitely often. All numerical evidence then available seemed to suggest that π(''x'') is always less than li(''x''), though mathematicians familiar with Riemann's work on the [[Riemann zeta function]] would probably have realized that occasional exceptions were likely by the argument [[#Riemann's formula\|given below]] (and the claim sometimes made that Littlewood's result was a big surprise to experts seems doubtful). Littlewood's proof did not, however, exhibit a ~~concrete such number ''x''~~.

	~~{{harvtxt\|Skewes\|1933}} proved that, assuming that the [[Riemann hypothesis]] is true, there exists~~ a ~~number ''x'' violating π(''x'') < li(''x'') below~~
	~~:<math>e^{e^{e^{79}}}<10^{10^{10^{34}}}.</math>~~

	~~In {{harv\|Skewes\|1955}}, without assuming the Riemann hypothesis, Skewes managed to prove that there must exist a value of ''x'' below~~
	~~:<math>e^{e^{e^{e^{7.705}}}}<10^{10^{10^{963}}}.</math>~~

	~~Skewes' task was to make Littlewood~~'s existence proof effective: exhibiting some concrete upper bound for the first sign change. According to [[George Kreisel]], this was at the time not considered obvious even in principle. The approach called ''[[Proof mining\|unwinding]]'' in [[proof theory]] looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

	~~Although both Skewes' numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as [[Graham's number]].~~

	~~==More recent estimates==~~
	~~These (enormous) 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 {{harvtxt\|Lehman\|1966}}, who~~
	~~showed that somewhere between 1.53{{e\|1165}} and 1.65{{e\|1165}} there are more than 10<sup>500</sup> consecutive integers ''x'' with π(''x'') > li(''x'').~~
	Without assuming the Riemann hypothesis, {{harvs\|txt=yes\|authorlink=Herman te Riele\|first=H. J. J.\|last= te Riele \|year= 1987}} proved an upper bound of 7{{e\|370}}. A better estimation was 1.39822{{e\|316}} discovered by {{harvtxt\|Bays\|Hudson\|2000}}, who showed there are at least 10<sup>153</sup> consecutive integers somewhere near this value where π(''x'') > li(''x''), and suggested that there are probably at least 10<sup>311</sup>. {{harvtxt\|Chao\|Plymen\|2010}} gave a small improvement and correction to the result of Bays and Hudson. Bays and Hudson found a few much smaller values of ''x'' where π(x) gets close to 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. {{harv\|Saouter\|Demichel\|2010}} find a smaller interval for a crossing, which was slightly improved by {{harv\|Zegowitz\|2010}}. The same source shows that there exists a number ''x'' violating π(''x'') < li(''x'') below <math>e^{727.951346801}</math>. The exponent could be reduced to 727.951338611, assuming Riemann hypothesis.

	~~Rigorously, {{harvtxt\|Rosser\|Schoenfeld\|1962}} proved that there are no crossover points below ''x'' = 10<sup>8</sup>, and~~ this ~~lower bound was subsequently improved by {{harvtxt\|Brent\|1975}} to 8{{e\|10}}, and by {{harvtxt\|Kotnik\|2008}} to 10<sup>14</sup>.~~

	~~There~~ is ~~no explicit value ''x'' known for certain to~~ have ~~the property π(x) > li(x), though computer calculations suggest some explicit numbers that are quite likely to satisfy this.~~

	{{harvtxt\|Wintner\|1941}} showed that the [[natural density\|proportion of integers]] for which π(''x'')-li(''x'') is positive, and {{harvtxt\|Rubinstein\|Sarnak\|1994}} showed that this proportion is about .00000026, which is surprisingly large given how far one has to go to find the first example.

	~~==Riemann's formula==~~
	~~Riemann gave an [[Explicit formulae (L-function)\|explicit formula]] for π(x), whose leading terms are (ignoring some subtle convergence questions)~~

	~~:<math>\pi(x) = \operatorname{li}(x) - \frac{\operatorname{li}(\sqrt{x})}{2} - \sum_\rho \operatorname{li}(x^\rho) + \text{smaller terms} </math>~~
	where the sum is over zeros ρ of the Riemann zeta function. The largest error term in the approximation π(''x'') = li(x) (if the [[Riemann hypothesis]] is true) is li({{radic\|''x''}})/2, showing that li(''x'') is usually larger than π(x). The other terms above are somewhat smaller, and moreover tend to have different complex arguments so mostly cancel out. Occasionally however, many of the larger ones might happen to have roughly the same complex argument, in which case they will reinforce each other instead of cancelling and will overwhelm the term li({{radic\|''x''}})/2. The reason why the Skewes number is so large is that these smaller terms are quite a lot smaller than the leading error term, mainly because the first complex zero of the zeta function has quite a large imaginary part, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term. The chance of ''N'' random complex numbers having roughly the same argument is about 1 in 2<sup>''N''</sup>. This explains why π(''x'') is sometimes larger than li(''x''), and also why it is rare for this to happen. It also shows why finding places where this happens depends on large scale calculations of millions of high precision zeros of the Riemann zeta function. The argument above is not a ~~proof, as~~ it ~~assumes the zeros of the Riemann zeta function are random which is not true~~. ~~Roughly speaking, Littlewood's proof consists of [[Dirichlet's approximation theorem]] to show that sometimes many terms have about~~ the ~~same argument.~~

	In the event that the Riemann hypothesis is false, the argument is much simpler, essentially because the terms li(''x''<sup>ρ</sup>) for zeros violating the Riemann hypothesis (with real part greater than 1/2) are eventually larger than li(''x''<sup>1/2</sup>).

	The reason for the term <math>\mathrm{li}(x^{1/2})/2</math> is that, roughly speaking, <math>\mathrm{li}(x)</math> counts not primes, but powers of primes <math>p^n</math> weighted by <math>1/n</math>, and <math>\mathrm{li}(x^{1/2})/2</math> is a sort of correction term coming from squares of primes.

	~~==References==~~
	~~{{reflist}}~~
	~~{{refbegin}}~~
	*{{citation\|mr=1752093\|first=C.\|last= Bays \|first2=R. H.\|last2= Hudson \|url=http://~~www~~.ams.org/mcom/2000-69-231/S0025-5718-99-01104-7/S0025-5718-99-01104-7.pdf \|title=A new bound for the smallest ''x'' with π(''x'') > li(''x'') \|journal=[[Mathematics of Computation]]\|volume=69\|year=2000 \|issue= 231\|pages= 1285–1296}}
	*{{citation\|mr=0369287\|first=R. P.\|last= Brent \|title=Irregularities in the distribution of primes and twin primes \|journal=[[Mathematics of Computation]]\|volume=29\|year=1975 \|pages= 43–56\|doi=10.2307/2005460\|jstor=2005460\|issue=129}}
	*{{citation\|unused_data=year=2010\|doi=10.1142/S1793042110003125\|title=A new bound for the smallest ''x'' with {{math\|π(''x'') > li(''x'')}}\|first=Kuok Fai\|last= Chao\|first2= Roger\|last2= Plymen\|year=2010\|arxiv=math/0509312 \|journal= International Journal of Number Theory \|volume= 6\|issue=03\|pages= 681–690\|mr=2652902}}
	*{{citation\|first= T.\|last= Kotnik \|doi=10.1007/s10444-007-9039-2 \|title=The prime-counting function and its analytic approximations \|journal=Advances in Computational Mathematics\|volume=29\|issue= 1\|year=2008\|pages= 55–70}}
	*{{citation\|first= R. Sherman \|last=Lehman\|title= On the difference π(''x'') − li(''x'')\|journal= [[Acta Arithmetica]] \|volume=11 \|year=1966\|pages= 397–410 \|mr=0202686}}
	* {{citation\|first=J. E.\|last= Littlewood\|title=Sur la distribution des nombres premiers\|journal=[[Comptes Rendus]]\|volume= 158 \|year=1914\|pages= 1869–1872}}
	*{{citation\|first= S.\|last= Skewes\|title=On the difference π(''x'') − Li(''x'')\|journal=[[Journal of the London Mathematical Society]]\|volume=8\|year=1933\|pages= 277–283}}
	*{{citation\|mr=0067145\| first= S.\|last= Skewes\|title=On the difference π(''x'') − Li(''x'') (II)\|journal=[[Proceedings of the London Mathematical Society]]\|volume= 5 \|year=1955\|pages= 48–70}}
	*{{citation\|mr=0866118\|first= H. J. J. \|last=te Riele\|title=On the sign of the difference π(''x'') − Li(''x'')\|journal=[[Mathematics of Computation]]\|volume=48\|year=1987\|pages= 323–328 \|jstor=2007893 \|issue=177 }}
	*{{citation\|mr=0137689\|first= J. B.\|last= Rosser \|first2= L.\|last2= Schoenfeld \|title=Approximate formulas for some functions of prime numbers \|journal=Illinois Journal of Mathematics\|volume=6\|year=1962\|pages= 64–94}}
	*{{citation
	~~\| last1 = Saouter \| first1 = Yannick~~
	~~\| last2 = Demichel \| first2 = Patrick~~
	~~\| doi = 10.1090/S0025-5718-10-02351-3~~
	~~\| mr = 2684372~~
	~~\| issue = 272~~
	~~\| journal = [[Mathematics of Computation]]~~
	~~\| pages = 2395–2405~~
	~~\| title = A sharp region where {{math\|π(''x'') − li(''x'')}} is positive~~
	~~\| volume = 79~~
	~~\| year = 2010}}~~
	*{{citation\|last=Zegowitz\|first=Stefanie\|title=On the positive region of <~~math~~>~~\pi(x)-\operatorname{li}(x)~~<~~/math~~>~~\|pages=69 pp.\|year=2010\|url=http://eprints.ma.man.ac.uk/1547/}}~~
	*{{citation\|mr=1329368 \|author2-link=Peter Sarnak \|last=Rubinstein\|first= M.~~\|last2= Sarnak\|first2= P. \|title=Chebyshev's bias~~
	~~\|journal=[[Experimental Mathematics (journal)\|Experimental Mathematics]] \|volume=3 \|year=1994\|issue= 3\|pages= 173–197 \|url= http://projecteuclid~~.~~org/euclid~~.~~em/1048515870 }}~~
	*{{citation\|mr=0004255\|last= Wintner\|first= A. \|title=On the distribution function of the remainder term of the prime number theorem \|journal= [[American Journal of Mathematics]]\|volume= 63\|year=1941\|pages= 233–248\|issue=2\|doi=10.2307/2371519\|jstor=2371519\|publisher=The Johns Hopkins University Press }}
	~~{{refend}}~~

	~~==External links==~~
	*[http://~~web.archive.org/web/20060908033007/http://demichel~~.net~~/patrick/li_crossover_pi.pdf The prime counting function and related subjects] by Patrick Demichels (retrieved 2009-09-29)~~

	~~{{Large numbers}}~~

	~~[[Category:Large numbers]]~~
	~~[[Category:Number theory]]~~
	~~[[Category:Large integers]~~]

Molecular clock - Revision history

en>AnomieBOT: Dating maintenance tags: {{Cit}}

en>Monkbot: /* Early discovery and genetic equidistance */Fix CS1 deprecated date parameter errors

en>RedBot: r2.7.2) (Robot: Adding nl:Moleculaire klok