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| In [[mathematics]], the '''Riemann–Siegel theta function''' is defined in terms of the [[Gamma function]] as
| | Let's have an appear on world's internet hosting business: Thousands and thousands of web hosts scattered around the world and their company aim is simple - to get recognized and make earnings. With so much sound around the internet hosting market, it is a hard job for us to select up the right hosting.<br><br>In any web business, we rely on excellent web hosts to provide steady and trustworthy webhosting services. Hostgator gives timely and promising qualities. They invested a great deal of cash for its servers which each of them utilizing Dual [https://Www.Gov.uk/search?q=Xeon+servers Xeon servers] with 4GB memory and IDE hard disk drives. These high speed servers provide high receptive uptime of up to 99.9 % without failure.<br><br>Hostgator shared hosting are charged $4. If you loved this article and you would like to get even more information concerning [http://turkishwiki.com/The_Stairs_Of_Network_Hosting_Arena_Registration Hostgator Voucher] kindly go to the webpage. 95 regular monthly for the fundamental shared hosting packaged. There still have a lot plans which offering cheaper rates, but if you begin to research and try Hostgator, you will feel that every dollar and cent you invested is simply not lost. Do your own research on the functions consisted of in every plans Hostgator providing, then you can [http://www.thefreedictionary.com/distinguish distinguish] the distinctions with others bundles offered by other internet hosts.<br><br>Hostgator likewise provides a bunch of other attributes. For exaple, Hostgator assists you to back up the internet websites promptly and securely. You have the choice to host unlimited domains, sub domains and FTP accounts. Of cause, the feature which mentioned just now will not be readily available if you are the standard hosting package subscriber. For standard package, Hostgator still offers you single domain which packaged together with twenty sub domains. In my viewpoint, you can begin your trial with Hostgator's standard plan. When business gets mature and thinking for expansion, you can anytime update your plan to more advance one.<br><br>Hostgator provides control panel. Out of so many web hosts around the world, Hostgator's cpanel can be said succeeding the leading ranking amongst them.<br><br>In regard to client supports, Hostgator has a really strong client care group which working 24/7 to help you tackle all kinds of issues relevant to their products. You can either contact them through phone or live chat with them with net. You can likewise email your issues to them. They will response immediately without taking any charges. This is why previously, Hostgator still the best option for customers.<br><br>A last words for those who are getting going - do not be scare by exactly what you are reviewing as building an excellent internet site is not that difficult. The work is in fact quite simple and direct as long as you're getting the right information. To obtain begun, you'll first require a domain name and a trustworthy hosting service. |
| | |
| :<math>\theta(t) = \arg \left( | |
| \Gamma\left(\frac{2it+1}{4}\right)
| |
| \right)
| |
| - \frac{\log \pi}{2} t</math>
| |
| | |
| for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and <math>\theta(0)=0</math> holds, i.e., in the same way that the [[principal branch]] of the [[logarithm|log]] Gamma function is defined.
| |
| | |
| It has an [[asymptotic expansion]]
| |
| | |
| :<math>\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots</math>
| |
| | |
| which is not convergent, but whose first few terms give a good approximation for <math>t \gg 1</math>. Its Taylor-series at 0 which converges for <math>|t| < 1/2</math> is
| |
| | |
| <math>\theta(t) = -\frac{t}{2} \log \pi + \sum_{k=0}^{\infty} \frac{(-1)^k \psi^{(2k)}\left(\frac{1}{4}\right) }{(2k+1)!} \left(\frac{t}{2}\right)^{2k+1}</math>
| |
| | |
| where <math>\psi^{(2k)}</math> denotes the [[Polygamma function]] of order <math>2k</math>.
| |
| The Riemann–Siegel theta function is of interest in studying the [[Riemann zeta function]], since it can rotate the Riemann zeta function such that it becomes the totally real valued [[Z function]] on the [[critical line]] <math>s = 1/2 + i t</math> .
| |
| | |
| == Curve discussion ==
| |
| | |
| The Riemann–Siegel theta function is an odd [[real analytic function]] for real values of ''t''. It has 3 roots at 0 and <math>\pm 17.8455995405\ldots</math> and it is an increasing function for values |''t''| > 6.29, because it has exactly one minima and one maxima at <math>\pm 6.289835988\ldots</math> with absolute value <math>
| |
| 3.530972829\ldots</math>. Lastly it has a unique inflection point at t=0 with <math>\theta^\prime(0)= -\frac{\ln \pi + \gamma + \pi/2 + 3 \ln 2}{2} = -2.6860917\ldots</math> where the theta function has its derivation minimum.
| |
| | |
| ==Theta as a function of a complex variable==
| |
| We have an infinite series expression for the log Gamma function
| |
| | |
| :<math>\log \Gamma \left(z\right) = -\gamma z -\log z | |
| + \sum_{n=1}^\infty
| |
| \left(\frac{z}{n} - \log \left(1+\frac{z}{n}\right)\right),</math>
| |
| | |
| where ''γ'' is [[Euler–Mascheroni constant|Euler's constant]]. Substituting <math>(2it+1)/4</math> for ''z'' and taking the imaginary part termwise gives the following series for ''θ''(''t'')
| |
| | |
| :<math>\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t
| |
| + \sum_{n=1}^\infty \left(\frac{t}{2n}
| |
| - \arctan\left(\frac{2t}{4n+1}\right)\right).</math>
| |
| | |
| For values with imaginary part between -1 and 1, the arctangent function is [[holomorphic function|holomorphic]], and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between -1/2 and 1/2, leading to a holomorphic function on this domain. It follows that the [[Z function]] is also holomorphic in this region, which is the critical strip.
| |
| | |
| We may use the identities
| |
| | |
| :<math>\arg z = \frac{\log z - \log\bar z}{2i}\quad\text{and}\quad\overline{\Gamma(z)}=\Gamma(\bar z)</math>
| |
| | |
| to obtain the closed-form expression
| |
| | |
| :<math>\theta(t) = \frac{\log\Gamma\left(\frac{2it+1}{4}\right)-\log\Gamma\left(\frac{-2it+1}{4}\right)}{2i} - \frac{\log \pi}{2} t,</math>
| |
| | |
| which extends our original definition to a holomorphic function of t. Since the principal branch of log Γ has a single branch cut along the negative real axis, θ(t) in this definition inherits branch cuts along the imaginary axis above i/2 and below -i/2.
| |
| | |
| {| style="text-align:center"
| |
| |+ '''Riemann–Siegel theta function in the complex plane'''
| |
| |[[Image:Riemann Siegel Theta 1.jpg|1000x140px|none]]
| |
| |[[Image:Riemann Siegel Theta 2.jpg|1000x140px|none]]
| |
| |[[Image:Riemann Siegel Theta 3.jpg|1000x140px|none]]
| |
| |-
| |
| |<math>
| |
| -1 < \Re(t) < 1
| |
| </math>
| |
| |<math>
| |
| -5 < \Re(t) < 5
| |
| </math>
| |
| |<math>
| |
| -40 < \Re(t) < 40
| |
| </math>
| |
| |}
| |
| | |
| ==Gram points==
| |
| The Riemann zeta function on the critical line can be written
| |
| | |
| :<math>\zeta\left(\frac{1}{2}+it\right) = e^{-i \theta(t)}Z(t),</math>
| |
| :<math>Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}+it\right).</math>
| |
| | |
| If '''<math>t</math>''' is a [[real number]], then the [[Z function]] <math>Z\left(t\right)</math> returns ''real'' values.
| |
| | |
| Hence the zeta function on the critical line will be ''real'' when
| |
| <math>\sin\left(\,\theta(t)\,\right)=0</math>. Positive real values of '''<math>t</math>''' where this occurs are called '''Gram points''', after [[Jørgen Pedersen Gram|J. P. Gram]], and can of course also be described as the points where <math>\frac{\theta(t)}{\pi}</math> is an integer.
| |
| | |
| A '''Gram point''' is a solution '''<math>g_{n}</math>''' of
| |
| :<math>\theta\left(g_{n}\right) = n\pi.</math> | |
| | |
| Here are the smallest non negative '''Gram points'''
| |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! <math>n</math>
| |
| ! <math>g_{n}</math>
| |
| ! <math>\theta(g_{n})</math>
| |
| |-
| |
| |style="text-align:right;"| -3
| |
| |style="text-align:right;"| 0
| |
| |style="text-align:right;"| 0
| |
| |-
| |
| |style="text-align:right;"| -2
| |
| |style="text-align:right;"| 3.4362182261...
| |
| |style="text-align:right;"| -π
| |
| |-
| |
| |style="text-align:right;"| -1
| |
| |style="text-align:right;"| 9.6669080561...
| |
| |style="text-align:right;"| -π
| |
| |-
| |
| |style="text-align:right;"| 0
| |
| | 17.8455995405...
| |
| |style="text-align:right;"| 0
| |
| |-
| |
| |style="text-align:right;"| 1
| |
| | 23.1702827012...
| |
| |style="text-align:right;"| π
| |
| |-
| |
| |style="text-align:right;"| 2
| |
| | 27.6701822178...
| |
| |style="text-align:right;"| 2π
| |
| |-
| |
| |style="text-align:right;"| 3
| |
| | 31.7179799547...
| |
| |style="text-align:right;"| 3π
| |
| |-
| |
| |style="text-align:right;"| 4
| |
| | 35.4671842971...
| |
| |style="text-align:right;"| 4π
| |
| |-
| |
| |style="text-align:right;"| 5
| |
| | 38.9992099640...
| |
| |style="text-align:right;"| 5π
| |
| |-
| |
| |style="text-align:right;"| 6
| |
| | 42.3635503920...
| |
| |style="text-align:right;"| 6π
| |
| |-
| |
| |style="text-align:right;"| 7
| |
| | 45.5930289815...
| |
| |style="text-align:right;"| 7π
| |
| |-
| |
| |style="text-align:right;"| 8
| |
| | 48.7107766217...
| |
| |style="text-align:right;"| 8π
| |
| |-
| |
| |style="text-align:right;"| 9
| |
| | 51.7338428133...
| |
| |style="text-align:right;"| 9π
| |
| |-
| |
| |style="text-align:right;"| 10
| |
| | 54.6752374468...
| |
| |style="text-align:right;"| 10π
| |
| |-
| |
| |style="text-align:right;"| 11
| |
| | 57.5451651795...
| |
| |style="text-align:right;"| 11π
| |
| |-
| |
| |style="text-align:right;"| 12
| |
| | 60.3518119691...
| |
| |style="text-align:right;"| 12π
| |
| |-
| |
| |style="text-align:right;"| 13
| |
| | 63.1018679824...
| |
| |style="text-align:right;"| 13π
| |
| |-
| |
| |style="text-align:right;"| 14
| |
| | 65.8008876380...
| |
| |style="text-align:right;"| 14π
| |
| |-
| |
| |style="text-align:right;"| 15
| |
| | 68.4535449175...
| |
| |style="text-align:right;"| 15π
| |
| |}
| |
| | |
| The choice of the index ''n'' is a bit crude. It is historically chosen in such a way that the index is 0 at the first value which is larger than the smallest positive zero (at imaginary part 14.13472515 ...) of the Riemann zeta function on the critical line. Notice, this <math>\theta</math>-function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [-24,24]! Thus the [[Odd function|odd]] theta-function has its symmetric Gram point with value 0 at index -3.
| |
| Gram points are useful when computing the zeros of <math>Z\left(t\right)</math>. At a Gram point '''<math>g_{n}</math>''',
| |
| | |
| :<math>\zeta\left(\frac{1}{2}+ig_n\right) = \cos(\theta(g_n))Z(g_n) = (-1)^n Z(g_n),</math>
| |
| | |
| and if this is ''positive'' at ''two'' successive Gram points, <math>Z\left(t\right)</math> must have a zero in the interval. | |
| | |
| According to '''Gram’s law''', the [[real part]] is ''usually'' positive while the [[imaginary part]] alternates with the '''gram points''', between ''positive'' and ''negative'' values at somewhat regular intervals.
| |
| | |
| :<math>(-1)^n \, Z\left(g_{n}\right) > 0</math>
| |
| | |
| The number of roots, <math>N\left(T\right)</math>, in the strip from 0 to ''T'', can be found by
| |
| :<math>N\left(T\right) = \frac{\theta(T)}{\pi} + 1+S(T),</math>
| |
| where <math>S(T)</math> is an error term which grows asymptotically like <math>\log T</math>.
| |
| | |
| Only if <math>g_{n}</math> '''would obey Gram’s law''', then finding the number of roots in the strip simply becomes
| |
| | |
| :<math>N\left(g_{n}\right) = n + 1.</math>
| |
| | |
| Today we know, that in the long run, '''Gram's law''' fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta-function. Gram was afraid that it may fail for larger indices (the first miss is at index 126 before the 127th zero) and thus claimed this only for not too high indices. Later Hutchinson coined the phrase ''Gram's law'' for the (false) statement that all zeroes on the critical line would be separated by Gram points.
| |
| | |
| ==See also==
| |
| *[[Z function]]
| |
| | |
| ==References==
| |
| *{{Citation | last1=Edwards | first1=H. M. | authorlink = Harold Edwards (mathematician) | title=Riemann's Zeta Function | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-41740-0 | mr=0466039 | year=1974}}
| |
| *{{citation|first=J. P.|last= Gram|title= Note sur les zéros de la fonction ζ(s) de Riemann|journal= Acta Mathematica|volume=27|issue=1|pages=289–304|year=1903|doi=10.1007/BF02421310}}
| |
| | |
| ==External links==
| |
| *{{MathWorld|title=Riemann-Siegel Functions|urlname=Riemann-SiegelFunctions}}
| |
| *[http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/RiemannSiegelTheta/ Wolfram Research – Riemann-Siegel Theta function] (includes function plotting and evaluation)
| |
| | |
| {{DEFAULTSORT:Riemann-Siegel theta function}}
| |
| [[Category:Zeta and L-functions]]
| |
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