Number of sign changes Pi(n) - Li(n) if RH is false

TauCeti · 2003-10-21, 15:43

I have one question regarding the prime counting function (Pi(n)) and the logarithmic integral (Li(n)).

At http://mathworld.wolfram.com/SkewesNumber.html there is an explanation, that Littlewood proved in 1914, that Pi(n)-Li(n) changes sign infinitely often. The earliest calculated upper bound for the first sign change was the famous first Skewes number. That upper bound (today improved) is valid if the RH is true.

There also exists a second Skewes Number that defines an upper bound for the first sign change if the RH is false.

Now my question: It the Littlewood proof of infinitely often sign changes only valid if the RH is true or isn't that a condition for that proof?

In other words: Is it proven, that Pi(n)-Li(n) changes sign more then once if RH is false?

I have tried to find references and even tried to access Littlewoods original paper on the net. But no success

So an answer to that question or a pointer to an explanation would really help me to improve my understanding of the topic.

Tau

philmoore · 2003-10-21, 16:16

I think I saw a reference to this in Davenport, "Multiplicative Number Theory", in which he says that Littlewood's proof contained two cases. Case 1 assumed that the Riemann hypothesis was true and case 2 assumed that the Riemann hypothesis was false. I think he also said that case 1 was the more difficult case, but I'm not sure. I'll check it tonight and give you a reference tomorrow.

philmoore · 2003-10-22, 18:20

I checked, and my memory was correct. The reference is Davenport, "Multiplicative Number Theory", chapter 30, "References to other work", page 172, 3rd edition:

"Littlewood's proof was divided into two cases, according as the Riemann hypothesis is true or false, the former being the difficult case. Owing to its indirect character the proof did not make it possible to name a particular number x_o such that pi(x) > li x for some x < x_o. It was not until 1955 that such a number was found, namely by Skewes: his number was 10^(10^(10^(10^3))). On the basis of extensive computer calculations, te Riele has shown that one may take x_o = 6.687x10^370.

However, Davenport does not cover the proof; he only gives references. I don't know if you can find any of these on the web, but Prachar's "Primzahlverteilung", Springer, 1957 is supposed to be one of the classics in this field. He references chapter 7, section 8 in connection to Littlewood's theorem. If any one is interested, I can also post references to the original articles as well.

TauCeti · 2003-10-23, 08:43

Thanks a lot for the references. With that information i found an interesting paper from Wolf (http://citeseer.nj.nec.com/91974.html) with additional references to my original question.

I guess now its time for me to visit the local university library to access the sources not available on the web.

P.S.: I hope you do not think i have gone nuts (regarding my thoughts about the RH)

Tau

2003-10-21, 15:43	#1
TauCeti Mar 2003 Braunschweig, Germany 2×113 Posts	Number of sign changes Pi(n) - Li(n) if RH is false I have one question regarding the prime counting function (Pi(n)) and the logarithmic integral (Li(n)). At http://mathworld.wolfram.com/SkewesNumber.html there is an explanation, that Littlewood proved in 1914, that Pi(n)-Li(n) changes sign infinitely often. The earliest calculated upper bound for the first sign change was the famous first Skewes number. That upper bound (today improved) is valid if the RH is true. There also exists a second Skewes Number that defines an upper bound for the first sign change if the RH is false. Now my question: It the Littlewood proof of infinitely often sign changes only valid if the RH is true or isn't that a condition for that proof? In other words: Is it proven, that Pi(n)-Li(n) changes sign more then once if RH is false? I have tried to find references and even tried to access Littlewoods original paper on the net. But no success So an answer to that question or a pointer to an explanation would really help me to improve my understanding of the topic. Tau

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2003-10-21, 16:16	#2
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 3·373 Posts	I think I saw a reference to this in Davenport, "Multiplicative Number Theory", in which he says that Littlewood's proof contained two cases. Case 1 assumed that the Riemann hypothesis was true and case 2 assumed that the Riemann hypothesis was false. I think he also said that case 1 was the more difficult case, but I'm not sure. I'll check it tonight and give you a reference tomorrow.

2003-10-22, 18:20	#3
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 3×373 Posts	I checked, and my memory was correct. The reference is Davenport, "Multiplicative Number Theory", chapter 30, "References to other work", page 172, 3rd edition: "Littlewood's proof was divided into two cases, according as the Riemann hypothesis is true or false, the former being the difficult case. Owing to its indirect character the proof did not make it possible to name a particular number x_o such that pi(x) > li x for some x < x_o. It was not until 1955 that such a number was found, namely by Skewes: his number was 10^(10^(10^(10^3))). On the basis of extensive computer calculations, te Riele has shown that one may take x_o = 6.687x10^370. However, Davenport does not cover the proof; he only gives references. I don't know if you can find any of these on the web, but Prachar's "Primzahlverteilung", Springer, 1957 is supposed to be one of the classics in this field. He references chapter 7, section 8 in connection to Littlewood's theorem. If any one is interested, I can also post references to the original articles as well.

2003-10-23, 08:43	#4
TauCeti Mar 2003 Braunschweig, Germany 2·113 Posts	Thanks a lot for the references. With that information i found an interesting paper from Wolf (http://citeseer.nj.nec.com/91974.html) with additional references to my original question. I guess now its time for me to visit the local university library to access the sources not available on the web. P.S.: I hope you do not think i have gone nuts (regarding my thoughts about the RH) Tau