Riemann–Siegel theta function

The Riemann - Siegel theta function is a special function from the analytic number theory, a branch of mathematics. It serves mainly the study of zeros of the Riemann zeta - function and thus. As a tool in the context of the Riemann Hypothesis, one to date unsolved problem in mathematics, whose solution would allow statements about the distribution of primes Thus, with the help of the Riemann - Siegel theta function to specify the number of so-called non-trivial zeros of the Riemann zeta function to a predetermined imaginary. The Riemann - Siegel theta function also appears in the definition of Gram- points - some real numbers, their location, the position of those zeros are often, but not always, confining.

The theta function is named after the two German mathematicians Bernhard Riemann and Carl Ludwig Siegel. Riemann, who died in 1866 at the age of 39 years, left numerous, individual worksheets and mathematical notes. Born in 1896 Siegel took on these documents and published in 1932 a work on Riemann's estate to analytic number theory. There he treated the today so called Riemann - Siegel formula and hence the theta function.

The illustrated in this article Riemann - Siegel theta function is to be distinguished from other mathematical functions, which also bear the name " theta function " such as the Jacobian or the Ramanujanschen theta function.

Definition

The Riemann - Siegel theta function is defined for real by

It refers to the circle number, the imaginary unit, the logarithm function, the gamma function and a function argument that is uniquely determined by the following conditions: the values ​​of the argument function must be determined so that the following applies and becomes a continuous function. [Note 1] [note 2] [note 3] [note 4]

This definition can also be in the form

Write if you choose for the main branch of the logarithm, and denotes the imaginary part of a complex number. [Note 5]

This last form of the definition is also to define the Riemann Siegel theta function of complex arguments:

Which is to choose the principal branch of the logarithm.

Properties of the theta function with a real argument

Curve Sketching

The Riemann - Siegel theta function with real argument is a real analytic function. In particular, it is continuous and infinitely differentiable. For example, as the sine function it belongs to the odd functions. It applies to everyone. In addition to 0, it still has the two zeros

The theta function with a real argument takes

A local minimum or maximum. Be the local function values

For positive, growing against it; negative, decreasing against.

Asymptotic Expansion

The Riemann - Siegel theta function with a real argument has an asymptotic expansion whose leading members have the following form:

In the derivation of this development are replaced in the definition of the function by the Stirling number and uses an identity between the logarithm of the complex and the arc tangent and the number representation. For larger values ​​of already after the limb cut off, asymptotic expansion provides good approximations to the actual values ​​of. It is therefore

If required, the quality of this approach can be increased further by means of further links in the asymptotic expansion.

Related to the Riemann zeta function

The Riemann zeta function is one of the most important functions of analytic number theory. Your paramount importance owes to the relationship between the position of their complex zeros and the distribution of primes. Your so-called trivial zeros it takes in the negative even numbers on, so in -2, -4, -6, -8, etc. In addition, it also possesses an infinite number of so-called non-trivial zeros, one of which is known, that their real parts lie between 0 and 1. Bernhard Riemann conjectured in his famous work of 1859, all non- trivial zeros of the zeta function possess the real part 1/2. This thesis, which is to this day neither proved nor disproved is called the Riemann Hypothesis.

In view of the Riemann Hypothesis is first attempted to obtain information about the zeros of the zeta function with real part 1/2. It turns out to be advantageous not to work directly with the Riemann zeta function, but with a close relative: the Riemann Xi function. This is for complex defined by

On the right side, the factors in front of the zeta function exactly eliminate the trivial zeros of the zeta function and the pole in 1 Thus, the zeros of the Xi - function are identical to the non- trivial zeros of the zeta function. Compared to the Riemann zeta function but the Xi - function has now the advantage, to take on the so-called critical line only real values ​​. Therefore, one can find simple zeros of the Xi - function with real part 1/2 and thus non-trivial zeros of the zeta function with real part 1/2, by examining to sign change along the critical line. It relies in. By simple transformations is then performed, not only the definition of the Riemann Siegel theta - function, but also on the definition of the Riemann Siegel Z function:

It denotes the real part of a complex number. As far as the question of the sign changes from along the critical line, so the expression evaluates within the first pair of square brackets for every real is always a negative real value. Further details of this expression will not be tested in the context of this sign changes. The function within the second pair of square brackets is exactly the Riemann - Siegel Z- function, which is named after the British mathematician Godfrey Hardy as Hardy's Z- function:

For further simplification are now pushing the two factors concerning the definition of another function, namely just the Riemann - Siegel theta function

Applies because with their help then

In summary, then:

When searching for sign changes of the Xi - function and thus by the zeros of the zeta function on the critical line, the last equation at first glance does not seem very profitable, because the value of is expressed with the help of. However, the value can be approximated by well without having to calculate the function value. This purpose is served the Riemann - Siegel formula

In which a natural number, and the cosine function respectively. In the Riemann - Siegel formula, the value depends on only and the selected, but no longer. This also applies to the error term. Using the theta function and the Riemann - Siegel formula therefore can be determined approximations of the values ​​of, and thus change in sign of. This change of sign indicate simple zeros of the Xi - function on the critical line and thus zeros of the zeta function with real part 1/2.

Number of non-trivial zeros of the Riemann zeta - function

Is a positive real number, then we denote by the number of all zeros of the Riemann zeta function with and. The value indicates the number of non- trivaler zeros of the zeta function with positive imaginary part in the so-called critical strip to which as the set of complex numbers is defined with real part [Note 6]. The zeros of counts according to their multiplicities, where so far only non-trivial zeros are simple multiplicity found.

Is now no imaginary part of a non-trivial zeros of the zeta function, so can be specified precisely using the Riemann - Siegel theta function and a error function of the value of:

Since the error function for -widening significantly slower growing than the value of approximately exactly the same.

Gram- points

The real zeros of the function are called Gram- points. So is a real number, it means a Gram- point, named after the Danish mathematician Jørgen Pedersen Gram. As sinusoidal function assumes its zero points in the integral multiple of, a real iff a Gram- point when

Valid for one.

Gram- points are usually numbered according to the following scheme: for the three real zeros of the theta function obviously Gram- points. The largest of these zeros is assigned the number 0 and decreases with, sometimes also referred to. Larger Gram- points are numbered in ascending order according to their size; smaller Gram- points in descending order. The table below shows the first non -negative Gram- points using these numbers:

If we now compare the Gram points with smaller number with the imaginary parts of the zeros of the Riemann zeta function along the critical line, then switch it off. The false thesis, for all Gram points alternated with the imaginary parts of these zeros from, was misleading by the American mathematician John Irwin Hutchinson as Grams referred law. The first of the infinitely many counterexamples to this " law " is found in the interval between the Gram- points

And

This interval does not contain an imaginary part of a zero of the Riemann zeta function. However, the following imaginary part of the zero- point of this interval just missed.