LOG#051. Zeta Zoology.Posted: 2012/11/07
This log-entry is an exploration journey… To boldly go, where no zeta function has gone before…
Riemann zeta function
The Riemann zeta function is an object related to prime numbers. In general, it is a function of complex variable defined by the next equation:
The Jacobi’s theta function is the Mellin transform of Riemann zeta function Jacobi theta function is
Applications: number theory, mathematics, physics, physmatics.
Related ideas: Hilbert-Polya approach, Riemann hypothesis, riemannium, primon gas/free Riemann gas, functional determinant, prime number distribution, Jacobi’s theta function.
Dirichlet eta function
This function is indeed the Riemann zeta function with alternating plus/minus signs. In other words:
Related ideas: Riemann zeta function.
Reciprocal Riemann zeta function
Reciprocal zeta function is the following modification of the Riemann zeta function:
where the Möbius function is defined as follows
A number is said to be square-free if it is not divisible by a number which is a perfect square (excepting the number one). An alternative definition of the Möbius function is given by:
and where is the number of different primes dividing the number and is the number of prime factors of , counted with multiplicities. Clearly, the inequality is satisfied. Moreover, note that and is undefined.
Indeed, we also have:
This result is important for the so-called Dirichlet generating series:
By the other hand, since
taking the ratio between these last two results, we obtain the beautiful equation
The Liouville function is defined similarly to the Möbius function. If is a positive integer, it is:
Using the sum of the geometric series, we get:
while if we use the Liouville function, we could write
There is other remarkable family of infinite products
where again counts the number of distinct prime factors of and is the number of square-free divisors. Furthermore, if is a Dirichlet character of conductor N, so that is totally multiplicative and only depends on , and if is not coprime to N, then the following identity holds
Here it is convenient and common to omit the primes dividing the conductor from the product.
Hurwitz zeta function
It is the the generalization of Riemann zeta function given by the next sum:
Remark: the mathematica code for this function is Zeta[s,Q].
Multiple zeta value/Euler sum/Polyzeta
Multiple zeta values, also called polyzeta function or Euler sums are certain “coloured” generalizations (in several variables) of the Riemann zeta function:
The polygogarithm is the following generalization of Riemann zeta function:
There are coloured versions of the polylogarithm:
The Lerch-zeta function is defined with the sum:
The Lerch trascendent is the function
Lerch-zeta function and Lerch trascendent are related through the functional equation
Mordell-Tornheim zeta values
Defined by Matsumoto in 2003, these zeta functions are:
Barnes zeta function
This function is the sum
where are numbers such that , and the sum is defined for all complex number s whenever .
Airy zeta function
Let be the zeros of the Airy function . Then, the Airy zeta function is the sum:
Arithmetic zeta function
The arithmetic zeta function over some scheme is defined to be the sum:
where the product is taken on every closed point of the scheme X.
The generalized Riemann hypothesis over the scheme is the hypothesis that the zeros of such arithmetic function, i.e, the feynmanity , and its poles are found in the next way:
inside the critical strip.
Artin-Mazur zeta function
Let us define:
1st. is the the set of fixed points of the nth iterated function of f.
2nd. is the cardinality of the set , i.e., the number of elements of such a set.
Then, the Artin-Mazur zeta function is the zeta function given by the next formula:
Dedekind zeta function
Let us define:
1st. is an algebraic number field.
2nd. is the range of non zero ideals of the ring of integers of K.
3rd. is the aboslute norm of I. When we get the usual Riemann zeta function.
Then, the Dedekind zeta function is the sum
Epstein zeta function/Eisenstein series
where we have defined as the quadratic form . A related concept is the Eisenstein (not confuse with Einstein, please)
where and the sum is taken on every pari of coprime integers. Two integers A and B are said to be coprime (also spelled co-prime) or relatively prime if the only positive integer that evenly divides both of them is 1.
There is a relation with modular forms/automorphic forms as well. Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the series:
It is absolutely convergent to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it can be extended to a holomorphic function at . It is a remarkable and surprising fact that the Eisenstein series is a modular form. Indeed, the key property is its -invariance. Explicitly if and then the next group property is satisfied
and is therefore a modular form of weight .
Remark: it is important to assume that otherwise it would be illegitimate to change the order of summation, and the -invariance would not remain. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for although it would only be what mathematicians call a quasimodular form.
Ihara zeta function
This zeta function appears in graph theory and it has an amazing set of useful identities. The Ihara zeta function is the sum:
where the product runs over every prime walk p of the graph , i.e., it is taken over closed cycles such as with and is equal to the length of the cycle p.
The Ihara formula is a key result in graph theory
and there is the circuit rank, i.e., it is the cyclomatic number of an undirected graph G or the minimum number of edges necessary to remove from G all its cycles, making it into a forest (graph without cycles, a fores is only a disjoint union of “trees”). Finally, if is the Hashimoto’s edge adjacency operator, then
Lefschetz zeta function
Given a map f, the Lefschetz zeta function is defined as the series
Here, is the Lefschetz number of the n-th iterated of the function f. To see what the Lefschetz number is, click here http://en.wikipedia.org/wiki/Lefschetz_number
Matsumoto zeta function
A class of zeta functions defined by Matsumoto around 1990. They are functions
where is a prime number and is certain polynomial.
Minakshisundaram-Pleijel zeta function
A type of zeta function encoding the eigenvalues of a Lapalacian of a compact riemannian manifold . If and the eigenvalues of the Laplace-Beltrami operator are the set , then the Minakshisundaram-Pleijel zeta function is defined as the following series (where we removed the zero eigenvalues from the sum and , i.e., the real part of s is large enough):
Prime Zeta function
The next function was defined by Fröberg, Cohen and Glaisher, with the only subtle point of being careful to consider as a prime in the sum or not and the notation they used:
Note that such a function is a “prime” version of the Riemann zeta function:
Remark: Cohen used a different notation for . He used instead of the Fröberg’s and Glaisher notation.
Remark (II): Interestingly, the prime zeta function has the following behaviour close to the axis
This prime zeta function is related to the Riemann zeta function:
This equation and definition can be inverted (the original inversion procedure was carried by Glaisher around 1891, it is recalled by Fröberg about 1968, and it was studied later by Cohen, circa 2000):
Remark: the mathematica code for the prime zeta function is PrimeZetaP[s] and Zeta[s] for the Riemann zeta function.
Remark (III): Fröberg (1968) stated that very little is known about the prime zeta function zeroes in the complex plane, i.e., the solutions to . Unlike the Riemann zeroes, it seems that prime zeta function zeroes are not on a straight line, but there is no known pattern, if any.
Remark (IV): Despite the divergence of , dropping the initial term and adding the Euler-Mascheroni constant provides a new constant! It is called Mertens constant. That is,
Remark (V): The Artins constant is related to as well
and where is the n-th Lucas number.
Remark (VI): The prime zeta function has the next asymptotical behaviour close to
Ruelle zeta function
Let’s define the following concepts:
1st. is certain function or map on a manifold M.
2nd. is the set of fixed points of the nth iterated function of f, being such an iterated function a finite value.
3rd. is certain function on M with values or entries in complex matrices. The case corresponds to the Artin-Mazur zeta function.
The Ruelle zeta function is the object defined with the series
Selberg zeta function
This zeta function is related to a compact ( of finite volume) Riemannian manifold. Assuming that certain manifolf M has constant curvature , it can be realized as a quotient of the Poincaré upper half plane
The Poincaré arc length is defined in this space as
and it can be shown to be invariant under fractional linear transformations
with and . Indeed, it is not hard to prove that the geodesics (curves minimizing the Poincaré arc length) are half lines and semicircles in H orthogonal to the real axis. Calling these lines as geodesics creates a model of hyperbolic geometry, i.e., a non-euclidean model for geometry where the 5th Euclid postulate is not longer valid. In fact, there are infinitely many geodesics through a fixed point not meeting a given geodesic. The fundamental group of M acts as a discrete group of transformations preserving distances between points. The favourite group between number theorists is called the modular group of matrices of determinant one and integer entries in the quotien space . However, the Riemann surface is noncompact, although it does have finite volume. Selberg introduced “prime numbers” in the compact surface to be “primitive cycles” or more precisely “primitive closed geodesics” C in M. There, the word “primitive” means that you can only go around the curve once. Furthermore, the Selberg zeta function, for large enough, is defined to be the sum
and where the product is extended over every primitive closed geodesics C in of Poincaré length . By the Selberg trace formula (which we are not goint to discuss here today), there is a duality between the lengths of the primes and the spectrum of the Laplace operator on M. Here, the Laplacian on M is
Indeed, it shows that one can show that the Riemann hypothesis (suitably modified to fit the situation) can be proved for Selberg zeta functions of compact Riemann surfaces! The closed geodesics in correspond to geodesics in H itself. One can show that the endpoints of such geodesics in the real line (note that the real line is the boundary of the set H) are fixed by hyperbolic elements of . That is, they are matrices
with trace . Primitive closed geodesics correspond to hyperbolic elements that generate their own centralizer in .
Shimizu zeta function
1st. , a totally algebraid number field.
2nd. , certain lattice in the field K.
3rd. , the subgroup of maximal rank of the group of the totally positive units preserving the lattice structure.
Then, the Shimizu zeta function arises in the form
Shintani zeta function
It is a generalized zeta series with the following formal definition
where are inhomogeneous functions of . Special cases of Shintani zeta function (or Shintani L-series, as they are also called by the mathematicians) are the Barnes zeta function or the Riemann zeta function.
Witten zeta function
Let G be a semisimple Lie group. The Witten L-series or Witten zeta function is defined by
This sum is taken over the equivalence classes of irreducible representations R of G. Considering a root system of rank equal to and with positive roots in , being all simple without loss of generality, the simple roots allow us to define the Witten zeta function as a function of several variables:
Zeta function of an operator
The zeta function of any (pseudo)-differential operator , or more generally any operator, can be defined as the following functional series:
and where the trace is taken over the values s where such number exists (i.e., the zero modes are removed). In fact, the zeta function of an arbitrary operator, that we can call the zetor, is the formal series:
It allow us to define the generalization of the determinant to -dimensional operators in the following non-trivial way:
They are the formal series
where is a Dirichlet character with conductor f, i.e.,
There, the generalized Bernoulli numbers are related to the L-series through the generating function above, and they satisfy the identity
p-adic zeta function
The p-adic analogue of the zeta function is defined with the following equation:
Moreover, we also define the zeta function at the infinite real prime:
The p-adic zeta function and the “real” prime zeta function (zeta function in the so-called “infinite prime”) satisfy the important adelic identity:
where , and is the classical Riemann zeta function. This adelic identity is just a special case of the adelic-type identity:
Stay tuned…The great adventure of Physmatics is just beginning!