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| In [[mathematics]], the '''explicit formulae for [[L-function]]s''' are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by {{harvtxt|Riemann|1859}} for the [[Riemann zeta function]]. Such explicit formulae have been applied also to questions on bounding the [[discriminant of an algebraic number field]], and the [[conductor of a number field]].
| | <br><br>It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you loved this post and you would certainly like to get even more facts regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] kindly check out our site. |
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| ==Riemann's explicit formula==
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| In his 1859 paper ''[[On the Number of Primes Less Than a Given Magnitude]]'' Riemann found an explicit formula for the normalized prime-counting function π<sub>0</sub>(''x'') which is related to the [[prime-counting function]] π(''x'') by
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| :<math>\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h)).</math>
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| His formula was given in terms of the related function
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| :<math>f(x) =\pi(x)+\frac{1}{2}\pi(x^{1/2})+\frac{1}{3}\pi(x^{1/3})+\cdots</math>
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| which counts primes where a prime power ''p''<sup>''n''</sup> counts as 1/''n'' of a prime and which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities. The normalized prime-counting function can be recovered from this function by
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| :<math>\pi_0(x) = \sum_n\mu(n)f(x^{1/n})/n = f(x) -\frac{1}{2}f(x^{1/2})-\frac{1}{3}f(x^{1/3}) - \cdots.</math>
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| Riemann's formula is then
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| :<math>f(x) = \operatorname{li}(x) - \sum_\rho \operatorname{li}(x^\rho) -\log(2) +\int_x^\infty\frac{dt}{t(t^2-1)\log(t)}</math> | |
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| involving a sum over the non-trivial zeros ρ of the Riemann zeta function. The sum is not [[Absolute convergence|absolutely convergent]], but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) [[logarithmic integral function]] given by the [[Cauchy principal value]] of the divergent integral
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| :<math>\operatorname{li}(x) = \int_0^x\frac{dt}{\log(t)}.</math>
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| The terms li(''x''<sup>ρ</sup>) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined by analytic continuation in the complex variable ρ in the region ''x''>1 and Re(ρ)>0. The other terms also correspond to zeros: the dominant term li(''x'') comes from the pole at ''s'' = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. (For graphs of the sums of the first few terms of this series see {{harvnb|Zagier|1977}}.)
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| A simpler variation of Riemann's formula using the normalization <math>\psi_0</math> of [[Chebyshev's function]] ψ rather than π is<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/ExplicitFormula.html Explicit Formula] on MathWorld.</ref> von-Mangoldt's explicit formula | |
| :<math>\psi_0(x) = \dfrac{1}{2\pi i}\int_0^{\infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds=x-\sum_\rho\frac{x^\rho}{\rho} - \log(2\pi) -\log(1-x^{-2})/2</math>
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| where for non-integral ''x'', ψ(''x'') is the sum of log(''p'') over all prime powers ''p''<sup>''n''</sup> less than ''x''. It plays an important role in von Mangoldt's proof of Riemann's explicit formula.
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| Here the sum over zeroes should again be taken in increasing order of imaginary part:<ref name=Ing77>Ingham (1990) p.77</ref>
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| :<math>\sum_\rho\frac{x^\rho}{\rho} = \lim_{T \rightarrow \infty} S(x,T) \ </math> | |
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| where
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| :<math>S(x,T) = \sum_{\rho:|\Im \rho| \le T} \frac{x^\rho}{\rho} \ . </math>
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| The error involved in truncating the sum to ''S''(''x'',''T'') is of order<ref name=Ing77/>
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| :<math> x^2 \frac{\log^2 T}{T} + \log x \ . </math>
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| ==Weil's explicit formula ==
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| There are several slightly different ways to state the explicit formula.
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| Weil's form of the explicit formula states
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| :<math>
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| \begin{align}
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| & {} \quad \Phi(1)+\Phi(0)-\sum_\rho\Phi(\rho) \\
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| & = \sum_{p,m} \frac{\log(p)}{p^{m/2}} (F(\log(p^m)) + F(-\log(p^m))) - \frac{1}{2\pi} \int_{-\infty}^\infty\varphi(t)\Psi(t)\,dt
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| \end{align}
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| </math>
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| where
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| *''ρ'' runs over the non-trivial zeros of the zeta function
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| *''p'' runs over positive primes
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| *''m'' runs over positive integers
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| *''F'' is a smooth function all of whose derivatives are rapidly decreasing
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| *<math>\varphi</math> is a Fourier transform of ''F'':
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| :: <math>\varphi(t) = \int_{-\infty}^\infty F(x)e^{itx}\,dx</math>
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| *<math>\Phi(1/2 + it) = \varphi(t)</math>
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| *<math>\Psi(t) = - \log( \pi ) + Re(\psi(1/4 + it/2))</math>, where <math>\psi</math> is the [[digamma function]] Γ<big>''′''</big>/Γ.
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| Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function is the set of prime powers plus some elementary factors.
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| The terms in the formula arise in the following way. | |
| *The terms on the right hand side come from the logarithmic derivative of
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| :: <math>\zeta^*(s)= \Gamma(s/2)\pi^{-s/2}\prod_p \frac{1}{1-p^{-s}}</math>
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| :with the terms corresponding to the prime ''p'' coming from the Euler factor of ''p'', and the term at the end involving ''Ψ'' coming from the gamma factor (the Euler factor at infinity). | |
| *The left-hand side is a sum over all zeros of ''ζ''<sup> *</sup> counted with multiplicities, so the poles at 0 and 1 are counted as zeros of order −1.
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| ==Generalizations==
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| The Riemann zeta function can be replaced by a [[Dirichlet L-function]] of a [[Dirichlet character]] χ. The sum over prime powers then gets extra
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| factors of χ(''p''<sup> ''m''</sup>), and the terms ''Φ''(0) and ''Φ''(0) disappear because the L-series has no poles.
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| More generally, the Riemann zeta function and the L-series can be replaced by the [[Dedekind zeta function]] of an algebraic number field or a [[Hecke L-series]]. The sum over primes then gets replaced by a sum over prime ideals.
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| ==Applications==
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| Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take ''F''(log(''y'')) to be ''y''<sup>1/2</sup>/log(''y'') for 0 ≤ ''y'' ≤ ''x'' and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than ''x''. The main term on the left is ''Φ''(1); which turns out to be the dominant terms of the [[prime number theorem]], and the main correction is the sum over non-trivial zeros of the zeta function. (There is a minor technical problem in using this case, in that the function ''F'' does not satisfy the smoothness condition.)
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| ==Hilbert–Pólya conjecture==
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| According to the [[Hilbert–Pólya conjecture]], the complex zeroes ρ should be the [[eigenvalue]]s of some [[linear operator]] ''T''. The sum over the zeros of the explicit formula is then (at least formally) given by a trace:
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| :<math> \sum_\rho F(\rho) = \operatorname{Tr}(F(\widehat T )).\!</math>
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| Development of the explicit formulae for a wide class of L-functions was given by {{harvtxt|Weil|1952}}, who first extended the idea to [[local zeta-function]]s, and formulated a version of a [[generalized Riemann hypothesis]] in this setting, as a positivity statement for a [[generalized function]] on a [[topological group]]. More recent work by [[Alain Connes]] has gone much further into the functional-analytic background, providing a trace formula the validity of which is equivalent to such a generalized Riemann hypothesis. A slightly different point of view was given by Ralf Meyer. Meyer has derived the explicit formula of Weil via harmonic analysis on adelic spaces.
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| ==See also==
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| *[[Selberg trace formula]]
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| ==References==
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| {{reflist}}
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| *{{Citation | authorlink=Albert Ingham | last1=Ingham | first1=A.E. | title=The Distribution of Prime Numbers | publisher=[[Cambridge University Press]] | isbn=978-0-521-39789-6 | mr=1074573 | year=1990 | zbl=0715.11045 | edition=2nd | origyear=1932 | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=30 | others=reissued with a foreword by [[Robert Charles Vaughan (mathematician)|R. C. Vaughan]] }}
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| *{{citation | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebraic number theory | edition=2nd ed. | series=Graduate Texts in Mathematics | volume=110 | location=New York, NY | publisher=[[Springer-Verlag]] | year=1994 | isbn=0-387-94225-4 | zbl=0811.11001 }}
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| *{{Citation | last1=Riemann | first1=Bernhard | title=Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse | url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ | year=1859 | journal=Monatsberichte der Berliner Akademie}}
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| *{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Sur les "formules explicites" de la théorie des nombres premiers | mr=0053152 | year=1952 | journal=Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] Suppl.-band M. Riesz | volume=1952 | pages=252–265 | zbl=0049.03205 | language=French }}
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| *{{Citation | last1 = Mangoldt | first1 = Hans von | title=Zu Riemanns Abhandlung "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" | journal = Journal für die reine und angewandte Mathematik | volume=114 | year=1895 | pages=255–305 | zbl=26.0215.03 | language=German | issn=0075-4102 }}
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| *{{Citation | last1 = Meyer | first1 = Ralf | title="On a representation of the idele class group related to primes and zeros of ''L''-functions" | journal = Duke Math. Journal | volume=127 | number=3 | year=2005 | pages=519–595 | zbl=1079.11044 | issn=0012-7094 }}
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| *{{citation | last1=Montgomery | first1=Hugh L. | author1-link=Hugh Montgomery (mathematician) | last2=Vaughan | first2=Robert C. | author2-link=Bob Vaughan | title=Multiplicative number theory. I. Classical theory | series=Cambridge Studies in Advanced Mathematics | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2007 | isbn=0-521-84903-9 | zbl=1142.11001 }}
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| *{{citation | last=Patterson | first=S.J. | title=An introduction to the theory of the Riemann zeta-function | series=Cambridge Studies in Advanced Mathematics | volume=14 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-33535-3 | zbl=0641.10029 }}
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| ==Further reading==
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| * {{citation | last=Edwards | first=H.M. | authorlink=Harold Edwards (mathematician) | title=Riemann's zeta function | series=Pure and Applied Mathematics | volume=58 | location=New York-London |publisher=Academic Press | year=1974 | isbn=0-12-232750-0 | zbl=0315.10035 }}
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| * {{citation | last=Riesel | first=Hans | authorlink=Hans Riesel | title=Prime numbers and computer methods for factorization | edition=2nd | series=Progress in Mathematics | volume=126 | location=Boston, MA | publisher=Birkhäuser | year=1994 | isbn=0-8176-3743-5 | zbl=0821.11001 }}
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| [[Category:Zeta and L-functions]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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