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In mathematics, the '''Hardy–Ramanujan theorem''', proved by {{harvtxt|Hardy|Ramanujan|1917}}, states  that the [[Normal order of an arithmetic function|normal order]] of the number ω(''n'') of distinct [[prime factor]]s of a number ''n'' is log(log(''n'')). Roughly speaking, this means that most numbers have about this number of distinct prime factors.
Home improvement can be an important part of your life because you are personalizing your home. You need to do all of your research so that you don't wind up losing a ton of money or even harming yourself or your loved ones. There are some effective ideas in this article to help you start.<br><br>You might only need to do something small to get more money out of your home when you sell it. If you are considering the sale of your home, think about making one or two little improvements. For instance, new paint works wonders for the interior and exterior of any home. This simple fix will cost only $100 or $200, but can increase your equity up to $1,000.<br><br>A drill is a necessary part of many renovations. You can make holes and drive screws of any size. It's best to have a cordless battery powered drill with at least 9 volts and 1/4, 1/8, 1/16, and 1/32 inch drill bits, and attachments for driving flat-head and Phillips screws.<br><br>If you're planning to use conventional wood in heating you house, understand that this might not be a good thing to do. Besides any consideration of the looks (and smells) of such a fireplace, you need to know that it is never that efficient. A great deal of the air it heats escapes directly up the chimney. In addition to heat loss up the chimney, fires require oxygen to burn, and that oxygen needs to come from within your home.<br><br>It's important that your bathroom has either a window or ventilation system. Showering and bathing lead to humidity, which leads to mold. Painting the bathroom is a good way to hide the mold but it will not treat the problem. You need to take precautionary measures to stop it from ever growing. Reduce the humidity in the space with ventilation or a window.<br><br>You don't have to replace your entire lock, you can swap out the cylinder instead. This part actually does the locking. If you lose keys or want to change locks, it is possible; just replace the cylinder. Should you prefer complete security changes, or to enhance your door, the entire unit should be replaced.<br><br>Although your floor in the kitchen looks level, you should still use a leveling tool when putting up new cabinets. Begin at your kitchen floor's highest point, and then place a line across the wall's length where your cabinets are going to be put in. This is needed to ensure that the counters are going to be level.<br><br>Before working in your bathroom or kitchen, shut the water off. It seems common sense, however many homeowners often make the error of overlooking this step and end up with major flooding, which actually requires more renovations.<br><br>Before you hire a contractor to remodel your home, you should be sure that he or she understands exactly what your budget limit is. If your contractor is aware of your budget, they will then be able to make decisions with that in mind. Also, this will help the contractors get better value for the money invested. This ensures that everyone wins.<br><br>While the reward is high, there is a lot of effort involved with making serious home improvements. Nevertheless, if you have the right information and can safely and properly renovate your house, you can have your dream home. So, help yourself by conducting research and using the advice above.<br><br>If you loved this article and you would like to get more info with regards to bathroom renovation ideas ([http://www.homeimprovementdaily.com http://www.homeimprovementdaily.com]) please visit the web site.
A more precise version states that for any real-valued function ψ(''n'') that tends to infinity as ''n'' tends to infinity
:<math>|\omega(n)-\log(\log(n))|<\psi(n)\sqrt{\log(\log(n))}</math>
or more traditionally
:<math>|\omega(n)-\log(\log(n))|<{(\log(\log(n)))}^{\frac12 +\varepsilon}</math>  
for ''[[almost all]]'' (all but an infinitesimal proportion of) integers. That is, let ''g''(''x'') be the number of positive integers ''n'' less than ''x'' for which the above inequality fails: then ''g''(''x'')/''x'' converges to zero as ''x'' goes to infinity.
 
A simple proof to the result {{harvtxt|Turán|1934}} was given by [[Pál Turán]], who proved that
 
:<math>\sum_{n \le x} | \omega(n) - \log\log n|^2 \ll x \log\log x \ . </math>
 
The same results are true of Ω(''n''), the number of prime factors of ''n'' counted with [[Multiplicity_(mathematics)#Multiplicity_of_a_prime_factor|multiplicity]].
This theorem is generalized by the [[Erdős–Kac theorem]], which shows that ω(''n'') is essentially [[Normal distribution|normally distributed]].
 
==References==
*{{citation|first=G. H.|last= Hardy| authorlink=G. H. Hardy| first2= S.|last2=Ramanujan| authorlink2=Srinivasa Ramanujan |title=The normal number of prime factors of a number ''n''| journal= Quarterly Journal of Mathematics |volume= 48  |year=1917|pages= 76–92 | url=http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper35/page1.htm | jfm=46.0262.03 }}
* {{citation | last1=Kuo | first1=Wentang | last2=Liu | first2=Yu-Ru | chapter=The Erdős–Kac theorem and its generalizations | pages=209-216 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=[[American Mathematical Society]] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1187.11024 }}
* {{citation | last=Turán | first=Pál | authorlink=Pál Turán | title=On a theorem of Hardy and Ramanujan | journal=Journal of the London Mathematical Society | volume=9 | pages=274-276 | year=1934 | issn=0024-6107 | zbl=0010.10401 }}
 
*{{springer|id=H/h110080|first=A.|last= Hildebrand}}
 
{{DEFAULTSORT:Hardy-Ramanujan theorem}}
[[Category:Analytic number theory]]
[[Category:Theorems about prime numbers]]

Latest revision as of 16:20, 5 January 2015

Home improvement can be an important part of your life because you are personalizing your home. You need to do all of your research so that you don't wind up losing a ton of money or even harming yourself or your loved ones. There are some effective ideas in this article to help you start.

You might only need to do something small to get more money out of your home when you sell it. If you are considering the sale of your home, think about making one or two little improvements. For instance, new paint works wonders for the interior and exterior of any home. This simple fix will cost only $100 or $200, but can increase your equity up to $1,000.

A drill is a necessary part of many renovations. You can make holes and drive screws of any size. It's best to have a cordless battery powered drill with at least 9 volts and 1/4, 1/8, 1/16, and 1/32 inch drill bits, and attachments for driving flat-head and Phillips screws.

If you're planning to use conventional wood in heating you house, understand that this might not be a good thing to do. Besides any consideration of the looks (and smells) of such a fireplace, you need to know that it is never that efficient. A great deal of the air it heats escapes directly up the chimney. In addition to heat loss up the chimney, fires require oxygen to burn, and that oxygen needs to come from within your home.

It's important that your bathroom has either a window or ventilation system. Showering and bathing lead to humidity, which leads to mold. Painting the bathroom is a good way to hide the mold but it will not treat the problem. You need to take precautionary measures to stop it from ever growing. Reduce the humidity in the space with ventilation or a window.

You don't have to replace your entire lock, you can swap out the cylinder instead. This part actually does the locking. If you lose keys or want to change locks, it is possible; just replace the cylinder. Should you prefer complete security changes, or to enhance your door, the entire unit should be replaced.

Although your floor in the kitchen looks level, you should still use a leveling tool when putting up new cabinets. Begin at your kitchen floor's highest point, and then place a line across the wall's length where your cabinets are going to be put in. This is needed to ensure that the counters are going to be level.

Before working in your bathroom or kitchen, shut the water off. It seems common sense, however many homeowners often make the error of overlooking this step and end up with major flooding, which actually requires more renovations.

Before you hire a contractor to remodel your home, you should be sure that he or she understands exactly what your budget limit is. If your contractor is aware of your budget, they will then be able to make decisions with that in mind. Also, this will help the contractors get better value for the money invested. This ensures that everyone wins.

While the reward is high, there is a lot of effort involved with making serious home improvements. Nevertheless, if you have the right information and can safely and properly renovate your house, you can have your dream home. So, help yourself by conducting research and using the advice above.

If you loved this article and you would like to get more info with regards to bathroom renovation ideas (http://www.homeimprovementdaily.com) please visit the web site.

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