Main Page

In mathematics, the Riemann–Siegel theta function is defined in terms of the Gamma function as

$\theta (t)=\arg \left(\Gamma \left({\frac {2it+1}{4}}\right)\right)-{\frac {\log \pi }{2}}t$

for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and $\theta (0)=0$ holds, i.e., in the same way that the principal branch of the log Gamma function is defined.

It has an asymptotic expansion

$\theta (t)\sim {\frac {t}{2}}\log {\frac {t}{2\pi }}-{\frac {t}{2}}-{\frac {\pi }{8}}+{\frac {1}{48t}}+{\frac {7}{5760t^{3}}}+\cdots$

which is not convergent, but whose first few terms give a good approximation for $t\gg 1$. Its Taylor-series at 0 which converges for $|t|<1/2$ is

$\theta (t)=-{\frac {t}{2}}\log \pi +\sum _{k=0}^{\infty }{\frac {(-1)^{k}\psi ^{(2k)}\left({\frac {1}{4}}\right)}{(2k+1)!}}\left({\frac {t}{2}}\right)^{2k+1}$

where $\psi ^{(2k)}$ denotes the Polygamma function of order $2k$. 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 $s=1/2+it$ .

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 $\pm 17.8455995405\ldots$ and it is an increasing function for values |t| > 6.29, because it has exactly one minima and one maxima at $\pm 6.289835988\ldots$ with absolute value $3.530972829\ldots$. Lastly it has a unique inflection point at t=0 with $\theta ^{\prime }(0)=-{\frac {\ln \pi +\gamma +\pi /2+3\ln 2}{2}}=-2.6860917\ldots$ 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

$\log \Gamma \left(z\right)=-\gamma z-\log z+\sum _{n=1}^{\infty }\left({\frac {z}{n}}-\log \left(1+{\frac {z}{n}}\right)\right),$

where γ is Euler's constant. Substituting $(2it+1)/4$ for z and taking the imaginary part termwise gives the following series for θ(t)

$\theta (t)=-{\frac {\gamma +\log \pi }{2}}t-\arctan 2t+\sum _{n=1}^{\infty }\left({\frac {t}{2n}}-\arctan \left({\frac {2t}{4n+1}}\right)\right).$

For values with imaginary part between -1 and 1, the arctangent function is 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

$\arg z={\frac {\log z-\log {\bar {z}}}{2i}}\quad {\text{and}}\quad {\overline {\Gamma (z)}}=\Gamma ({\bar {z}})$

to obtain the closed-form expression

$\theta (t)={\frac {\log \Gamma \left({\frac {2it+1}{4}}\right)-\log \Gamma \left({\frac {-2it+1}{4}}\right)}{2i}}-{\frac {\log \pi }{2}}t,$

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.

Riemann–Siegel theta function in the complex plane

$-1<\Re (t)<1$	$-5<\Re (t)<5$	$-40<\Re (t)<40$

Gram points

The Riemann zeta function on the critical line can be written

$\zeta \left({\frac {1}{2}}+it\right)=e^{-i\theta (t)}Z(t),$

$Z(t)=e^{i\theta (t)}\zeta \left({\frac {1}{2}}+it\right).$

If $t$ is a real number, then the Z function $Z\left(t\right)$ returns real values.

Hence the zeta function on the critical line will be real when $\sin \left(\,\theta (t)\,\right)=0$. Positive real values of $t$ where this occurs are called Gram points, after J. P. Gram, and can of course also be described as the points where ${\frac {\theta (t)}{\pi }}$ is an integer.

A Gram point is a solution $g_{n}$ of

$\theta \left(g_{n}\right)=n\pi .$

Here are the smallest non negative Gram points

$n$	$g_{n}$	$\theta (g_{n})$
-3	0	0
-2	3.4362182261...	-π
-1	9.6669080561...	-π
0	17.8455995405...	0
1	23.1702827012...	π
2	27.6701822178...	2π
3	31.7179799547...	3π
4	35.4671842971...	4π
5	38.9992099640...	5π
6	42.3635503920...	6π
7	45.5930289815...	7π
8	48.7107766217...	8π
9	51.7338428133...	9π
10	54.6752374468...	10π
11	57.5451651795...	11π
12	60.3518119691...	12π
13	63.1018679824...	13π
14	65.8008876380...	14π
15	68.4535449175...	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 $\theta$-function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [-24,24]! Thus the odd theta-function has its symmetric Gram point with value 0 at index -3. Gram points are useful when computing the zeros of $Z\left(t\right)$. At a Gram point $g_{n}$,

$\zeta \left({\frac {1}{2}}+ig_{n}\right)=\cos(\theta (g_{n}))Z(g_{n})=(-1)^{n}Z(g_{n}),$

and if this is positive at two successive Gram points, $Z\left(t\right)$ 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.

$(-1)^{n}\,Z\left(g_{n}\right)>0$

The number of roots, $N\left(T\right)$, in the strip from 0 to T, can be found by

$N\left(T\right)={\frac {\theta (T)}{\pi }}+1+S(T),$

where $S(T)$ is an error term which grows asymptotically like $\log T$.

Only if $g_{n}$ would obey Gram’s law, then finding the number of roots in the strip simply becomes

$N\left(g_{n}\right)=n+1.$

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

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
• Wolfram Research – Riemann-Siegel Theta function (includes function plotting and evaluation)