LOG#051. Zeta Zoology.

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:

\boxed{\displaystyle{\zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}\dfrac{1}{n^s}=\prod_{p=2}^{\infty}\dfrac{1}{1-p^{-s}}=\prod_{p}\dfrac{1}{1-p^{-s}}}}

or

\boxed{\displaystyle{\zeta (s)=\dfrac{1}{1-2^{-s}}\dfrac{1}{1-3^{-s}}\ldots\dfrac{1}{1-137^{-s}}\ldots}}

The Jacobi’s theta function is the Mellin transform of Riemann zeta function Jacobi theta function is

\boxed{\displaystyle{\theta (\tau)=\sum_{n=-\infty}^{\infty}e^{\pi i n^2\tau}}}

and then

\boxed{\displaystyle{\zeta (s)=\dfrac{\pi^{s/2}}{2\Gamma (\frac{s}{2})}\int_0^\infty \theta (it)t^{s/2-1}dt}}

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:

\boxed{\displaystyle{\eta (s)=\sum_{n=1}^{\infty}(-1)^{n+1}n^{-s}=\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n^s}=\left(1-2^{1-s}\right)\zeta (s)}}

Applications: physmatics.

Related ideas: Riemann zeta function.

Reciprocal Riemann zeta function

Reciprocal zeta function is the following modification of the Riemann zeta function:

\boxed{\displaystyle{\dfrac{1}{\zeta (s)}=\sum_{n=1}^{\infty}\mu (n)n^{-s}=\sum_{n=1}^{\infty}\dfrac{\mu (n)}{n^s}}}

where the Möbius function \mu (n) is defined as follows

\mu (n)=\begin{cases}1\;\; \mbox{if n is a square-free positive integer with even number of prime factors}\\ -1\;\; \mbox{if n is a square-free positive integer with odd number of prime factors}\\ 0\;\; \mbox{if n is not square-free }\end{cases}

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:

\mu (n)=\begin{cases}(-1)^{\omega (n)}=(-1)^{\Omega (n)}\;\; \mbox{if}\;\;\omega (n)=\Omega (n)\\ 0\;\;\mbox{if}\;\;\omega (n)<\Omega (n)\end{cases}

and where \omega (n) is the number of different primes dividing the number n and \Omega (n) is the number of prime factors of n, counted with multiplicities. Clearly, the inequality \omega (n)\leq \Omega (n) is satisfied. Moreover, note that \mu (1)=1 and \mu (0) is undefined.

Indeed, we also have:

\boxed{\displaystyle{\dfrac{1}{\zeta (s)}=\left( \prod_p^\infty \dfrac{1}{1-p^{-s}} \right)^{-1}=\prod_p^\infty \left( 1-\dfrac{1}{p^s}\right)}}

This result is important for the so-called Dirichlet generating series:

\boxed{\displaystyle{\dfrac{\zeta (s)}{\zeta (2s)}=\sum_{n=1}^{\infty} \dfrac{\vert\mu (n)\vert }{n^{s}}=\prod_p^\infty \left(1+p^{-s}\right)}}

By the other hand, since

\boxed{\displaystyle{\dfrac{1}{\zeta(s)}=\prod_{p}^\infty (1-p^{-s}) = \sum_{n=1}^{\infty} \dfrac{\mu (n)}{n^{s}}}}

taking the ratio between these last two results, we obtain the beautiful equation

\boxed{\displaystyle{\dfrac{\zeta(s)^2}{\zeta(2s)}=\prod_{p} \left(\dfrac{1+p^{-s}}{1-p^{-s}}\right) = \prod_{p} \left(\dfrac{p^{s}+1}{p^{s}-1}\right)}}

The Liouville function \lambda (n) is defined similarly to the Möbius function. If n is a positive integer, it  is:

\lambda (n)=(-1)^{\Omega (n)}

Using the sum of the geometric series, we get:

\boxed{\displaystyle{\zeta(s)=\prod_{p} (1-p^{-s})^{-1}=\prod_{p} \left(\sum_{n=0}^{\infty}p^{-ns}\right) =\sum_{n=1}^{\infty} \dfrac{1}{n^{s}}}}

while if we use the Liouville function, we could write

\boxed{\displaystyle{\dfrac{\zeta(2s)}{\zeta(s)}=\prod_{p} (1+p^{-s})^{-1} = \sum_{n=1}^{\infty} \frac{\lambda(n)}{n^{s}}}}

There is other remarkable family of infinite products

\boxed{\displaystyle{\prod_{p} (1+2p^{-s}+2p^{-2s}+\cdots) = \sum_{n=1}^{\infty}2^{\omega(n)} n^{-s} = \dfrac{\zeta(s)^2}{\zeta(2s)}}}

where again \omega(n) counts the number of distinct prime factors of n and 2^{\omega(n)} is the number of square-free divisors. Furthermore,  if \chi (n) is a Dirichlet character of conductor N, so that \chi is totally multiplicative and \chi (n) only depends on n \;(mod N), and \chi (n)=0  if n is not coprime to N, then the following identity holds

\boxed{\displaystyle{\prod_{p} (1- \chi(p) p^{-s})^{-1} = \sum_{n=1}^{\infty}\chi(n)n^{-s}}}

Here it is convenient and common to omit the primes p dividing the conductor N from the product.

Hurwitz zeta function

It is the the generalization of Riemann zeta function given by the next sum:

\boxed{\displaystyle{\zeta (s,Q)=\sum_{n=0}^{\infty}\dfrac{1}{\left(Q+n\right)^{s}}=\dfrac{1}{Q^s}+\sum_{n=1}^{\infty}\dfrac{1}{\left(Q+n\right)^{s}}}}

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:

\boxed{\displaystyle{\zeta (s_1,s_2,\ldots,s_m)=\sum_{n_1>n_2>\ldots>n_m>0}^\infty\dfrac{1}{n_1^{s_1}n_2^{s_2}\cdots n_m^{s_m}}=\sum_{n_1>n_2>\ldots>n_m>0}^\infty \prod_{j=1}^m \dfrac{1}{n_j^{s_j}}}}

Polylogarithm/Coloured polylogarithm

The polygogarithm is the following generalization of Riemann zeta function:

\boxed{\displaystyle{\mbox{Li}_s (z)=\sum_{n=1}^{\infty}\dfrac{z^n}{n^s}=\sum_{n=1}^{\infty}z^n n^{-s}}}

There are coloured versions of the polylogarithm:

\boxed{\displaystyle{\mbox{Li}_{ (s_1,s_2,\ldots,s_m) }(z_1,z_2,\ldots,z_m)=\sum_{n_1>n_2>\ldots>n_m>0}^\infty\dfrac{z_1^{s_1}z_2^{s_2}\cdots z_m^{s_m}}{n_1^{s_1}n_2^{s_2}\cdots n_m^{s_m}}=\sum_{n_1>n_2>\ldots>n_m>0}^\infty \prod_{j=1}^m \dfrac{z_j^{s_j}}{n_j^{s_j}}}}

Lerch-zeta function/Lerch-trascendent

The Lerch-zeta function is defined with the sum:

\boxed{\displaystyle{L(\lambda, Q,s)=\sum_{n=0}^{\infty}\dfrac{e^{2\pi i \lambda n}}{(n+Q)^s}}}

The Lerch trascendent is the function

\boxed{\displaystyle{\Phi (z,s,Q)=\sum_{n=0}^{\infty}\dfrac{z^n}{(n+Q)^s}}}

Lerch-zeta function and Lerch trascendent are related through the functional equation

\Phi ( e^{2\pi i\lambda},s,Q)=L( \lambda ,Q,s)

Mordell-Tornheim zeta values

Defined by Matsumoto in 2003, these zeta functions are:

\boxed{\displaystyle{\zeta_{MT,r} (s_1,s_2,\ldots,s_r; s_{r+1})=\sum_{m_1>\cdots>m_r>0}\dfrac{1}{m_1^{s_1}\cdots m_r^{s_r} (m_1+\ldots+m_r )^{s_{r+1}}}}}

Barnes zeta function

This function is the sum

\boxed{\displaystyle{\zeta_N ( s,\omega\vert a_1,\ldots,a_N)=\sum_{n_1\ldots n_N\geq 0}\dfrac{1}{\left(\omega+n_1a_1+\cdots+n_N a_N\right)^s}}}

where \omega, a_j are numbers such that Re(\omega)>0, Re(a_j)>0 and the sum is defined for all complex number s whenever Re(s)>N.

Airy zeta function

Let  a_i \forall i=1,2,\ldots,\infty be the zeros of the Airy function \mbox{Ai} (x). Then, the Airy zeta function is the sum:

\boxed{\displaystyle{\zeta_{Ai} (s)=\sum_{i=1}^{\infty}\dfrac{1}{\vert a_i\vert ^s}}}

Arithmetic zeta function

The arithmetic zeta function over some scheme X is defined to be the sum:

\displaystyle{\zeta _X (s)=\prod_x \left(1-N(x)^{-s}\right)^{-1}}

where the product is taken on every closed point of the scheme X.

The generalized Riemann hypothesis over the scheme X is the hypothesis that the zeros of such arithmetic function, i.e, the feynmanity \zeta_X (s)=0, and its poles are found in the next way:

\boxed{\zeta_X (s)=0\leftrightarrow \begin{cases}\mbox{Zeroes at}\;\;\mbox{Re}(s)=\dfrac{1}{2},\dfrac{3}{2},\ldots,\infty\\ \mbox{Poles at}\;\;\mbox{Re}(s)=0,1,2,\ldots,\infty\end{cases}}

inside the critical strip.

Artin-Mazur zeta function

Let us define:

1st. \mbox{Fix}(f^n) is the the set of fixed points of the nth iterated function f^n of f.

2nd. \mbox{Card(Fix)}(f^n) is the cardinality of the set \mbox{Fix}(f^n), 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:

\boxed{\displaystyle{\zeta_f (s)=\exp \left(\sum_{n=1}^{\infty} \mbox{Card(Fix)}\left[ f^n\right]\dfrac{z^n}{n}\right)}}

Dedekind zeta function

Let us define:

1st. K is an algebraic number field.

2nd. I is the range of non zero ideals of the ring of integers \mathcal{O}_K of K.

3rd. N_{K/Q} (I) is the aboslute norm of I. When K=\mathbb{Q} we get the usual Riemann zeta function.

Then, the Dedekind zeta function is the sum

\boxed{\displaystyle{\zeta_K (s)=\sum_{I\subseteq \mathcal{O}_K}\dfrac{1}{\left(N_{K/\mathbb{Q}}(I)\right)^s}}}

where \mbox{Re}(s)>1.

Epstein zeta function/Eisenstein series

\boxed{\displaystyle{\zeta_Q (s)=\sum_{(m,n)\neq (0,0)}\dfrac{1}{Q(m,n)^s}}}

where we have defined Q(m,n) as the quadratic form Q(m,n)=cm^2+bmn+an^2. A related concept is the Eisenstein (not confuse with Einstein, please)

\boxed{\displaystyle{E(z,s)=\dfrac{1}{2}\sum_{(m,n)=1}\dfrac{y^s}{\vert mz+n\vert^{2s}}}}

where \mbox{Re}(s)>1 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 \tau be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series  G_{2k}(\tau)  of weight 2k,  where k\geq 2 is an integer, by the  series:

\boxed{\displaystyle{G_{2k}(\tau) = \sum_{ (m,n)\in\mathbb{Z}^2\backslash(0,0)} \dfrac{1}{(m+n\tau )^{2k}}}}

It is absolutely convergent to a holomorphic function of \tau in the upper half-plane and its Fourier expansion given below shows that it can be extended to a holomorphic function at \tau=i\infty. It is a remarkable and surprising fact that the Eisenstein series is a modular form. Indeed, the key property is its SL_2(\mathbb{Z})-invariance. Explicitly if a,b,c,d \in \mathbb{Z} and ad-bc=1 then the next group property is satisfied

\displaystyle{G_{2k} \left( \dfrac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)}

and G_{2k} is therefore a modular form of weight 2k.

Remark:  it is important to assume that k\geq 2, otherwise it would be illegitimate to change the order of summation, and the SL_2(\mathbb{Z})-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 k=1, 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:

\boxed{\displaystyle{\zeta_G (u)=\prod_{p}\left( 1-u^{L(p)}\right)^{-1}}}

where the product runs over every prime walk p of the graph G(E,V), i.e., it is taken over closed cycles p=(u_0,u_1,\ldots,u_{L(p)-1};u_0) such as (u_i,u_{(i+2)\mbox{mod}}\; L(p))\in E with u_i\neq u_{(i+2)\mbox{mod}\;L(p)} and L(p) is equal to the length of the cycle p.

The Ihara formula is a key result in graph theory

\boxed{\zeta_G (u)=\dfrac{\left(1-u^2\right)^{\chi (G)}}{\det \left(I-Au-(k-1)u^2(I)\right)}}

and there \chi (G) is the circuit rank, i.e., it is the cyclomatic number of an undirected graph G or  the minimum number r 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 T is the Hashimoto’s edge adjacency operator, then

\boxed{\displaystyle{\zeta_G (u)=\dfrac{1}{\det (1-Tu)}}}

Lefschetz zeta function

Given a map f, the Lefschetz zeta function is defined as the series

\boxed{\displaystyle{\zeta_f (s)=\exp \left[\sum_{n=1}^\infty L(f^n)\dfrac{z^n}{n}\right]}}

Here, L(f^n) is the Lefschetz number of the n-th iterated f^n 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

\boxed{\displaystyle{\phi (s)=\prod_p\dfrac{1}{A_p (p^{-s})}}}

where p is a prime number and A_p is certain polynomial.

Minakshisundaram-Pleijel zeta function

A type of zeta function encoding the eigenvalues of a Lapalacian of a compact riemannian manifold \mathcal{M}. If \mbox{dim}\mathcal{M}=N and the eigenvalues of the Laplace-Beltrami operator are the set \left(\lambda_1,\lambda_2,\ldots\right), then the Minakshisundaram-Pleijel zeta function is defined as the following series (where we removed the zero eigenvalues from the sum and \mbox{Re}(s)>>1, i.e., the real part of s is large enough):

\boxed{\displaystyle{\mathcal{Z} (s)=\mbox{Tr}(A^{-s})=\sum_{n=1, \lambda_n\neq 0}^\infty \vert \lambda_n\vert^{-s}}}

Prime Zeta function

The next function was defined by Fröberg, Cohen and Glaisher, with the only subtle point of being careful to consider 1 as a prime in the sum or not and the notation they used:

\boxed{\displaystyle{P(s)=\sum_p\dfrac{1}{p^s}=\sum_p p^{-s}}}

Note that such a function is a “prime” version of the Riemann zeta function:

\displaystyle{\zeta (s)=\sum_{k=1}^\infty k^{-s}}

Remark: Cohen used a different notation for P(s). He used P(s)=S_s instead of the Fröberg’s and Glaisher notation.

Remark (II): Interestingly, the prime zeta function has the following behaviour close to the axis s=1

P(1+\varepsilon)=-\ln \varepsilon+C+\mathcal{O}(\varepsilon)

where

\displaystyle{C=\sum_{n=2}^\infty \dfrac{\mu (n)}{n}\ln \zeta (n)\approx -0.315718452\ldots}

This prime zeta function is related to the Riemann zeta function:

\displaystyle{\ln \zeta (s)=-\sum_{p\geq 2}\ln \left(1-p^{-s}\right)=\sum_{p\geq 2}\sum_{k=1}^\infty\dfrac{p^{-ks}}{k}}

so

\boxed{\displaystyle{\ln \zeta (s)=\sum_{k=1}^\infty\dfrac{1}{k}\sum_{p\geq 2}p^{-ks}=\sum_{k=1}^\infty\dfrac{P(ks)}{k}=\sum_{n>0}\dfrac{P(ns)}{n}=\sum_{n=1}^\infty\dfrac{P(ns)}{n}}}

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):

\boxed{\displaystyle{P(s)=\sum_{k=1}^\infty \dfrac{\mu (k)}{k}\ln \left( \zeta (ks)\right)}}

Remark: the mathematica code for the prime zeta function is PrimeZetaP[s] and Zeta[s] for the Riemann zeta function.

Remark (II): \displaystyle{P(1)=\sum \dfrac{1}{p}=\infty}

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 P(s)=0. 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 P(1), dropping the initial term and adding the Euler-Mascheroni constant \gamma_E\approx 0.577\cdots provides a new constant! It is called Mertens constant. That is,

\displaystyle{\mbox{MERTENS CONSTANT}=B_1=\gamma_E+\sum_{m=2}^\infty\dfrac{\mu (m)}{m}\ln \left(\zeta (m)\right)\approx 0.2614972128\ldots}

Remark (V): The Artins constant C_{A} is related to P(n) as well

\displaystyle{\ln C_A=-\sum_{n=2}^\infty \dfrac{(L_n-1)P(n)}{n}}

and where L_n is the n-th Lucas number.

Remark (VI): The prime zeta function has the next asymptotical behaviour close to s=1

P(s)\rightarrow P(s)\approx \ln \zeta (s)\sim \ln \left(\dfrac{1}{s-1}\right)

Ruelle zeta function

Let’s define the following concepts:

1st. f is certain function or map on a manifold M.

2nd. \mbox{Fix}(f^n) is the set of fixed points of the nth iterated function f^n of f, being such an iterated function a finite value.

3rd. \phi is certain function on M with values or entries in d\times d complex matrices. The case d=1, \phi=1 corresponds to the Artin-Mazur zeta function.

The Ruelle zeta function is the object defined with the series

\boxed{\displaystyle{\zeta (z)=\exp \left(\sum_{m>\geq 1}\dfrac{z^m}{m}\sum_{x\in \mbox{Fix}(f^m)}\mbox{Tr}\left(\prod_{k=0}^{m-1}\phi \left[ f^k(x)\right]\right)\right)}}

Selberg zeta function

This zeta function is related to a compact ( of finite volume) Riemannian manifold. Assuming that certain manifolf M has constant curvature -1, it can be realized as a quotient of the Poincaré upper half plane

H=\{x+iy\vert x, y\in \mathbb{R},y>0\}

The Poincaré arc length is defined in this space as

ds^2=\dfrac{dx^2+dy^2}{y^2}

and it can be shown to be invariant under fractional linear transformations

z\rightarrow z'=\dfrac{az+b}{cz+d}

with a,b,c,d\in \mathbb{R} and ad-bc>0. 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 \Gamma of M acts as a discrete group of transformations preserving distances between points. The favourite group between number theorists is called the modular group \Gamma =SL(2,\mathbb{Z}) of 2\times 2 matrices of determinant one and integer entries in the quotien space \overline{\Gamma}=\Gamma/\{\pm I\}. However, the Riemann surface M=SL(2,\mathbb{Z})/H is noncompact, although it does have finite volume. Selberg introduced “prime numbers” in the compact surface M=\Gamma/H 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 \text{Re} (s) large enough, is defined to be the sum

\boxed{\displaystyle{Z(s)=\prod_{\left[C\right]}\prod_{j\geq 1}\left(1-e^{(s+j)\nu (C)}\right)}}

and where the product is extended over every primitive closed geodesics C in M=\Gamma/H of Poincaré length \nu (C). 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

\Delta =y^2\left(\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial ^2}{\partial y^2}\right)

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 M=\Gamma/H correspond to geodesics in H itself. One can show that the endpoints of such geodesics in the real line \mathbb{R} (note that the real line is the boundary of the set H) are fixed by hyperbolic elements of \Gamma. That is, they are matrices

\begin{pmatrix} a & b\\ c & d\end{pmatrix}

with trace a+d>2. Primitive closed geodesics correspond to hyperbolic elements that generate their own centralizer in \Gamma.

Shimizu zeta function

We define:

1st. K, a totally algebraid number field.

2nd. M, certain lattice in the field K.

3rd. V, 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

\boxed{\displaystyle{L(M,V,s)=\sum_{p\in \left[M-0\right]}\dfrac{\mbox{sign}N(\mu)}{\vert N(\mu)\vert^s}}}

Shintani zeta function

It is a generalized zeta series with the following formal definition

\boxed{\displaystyle{\zeta (s_1,s_2,\ldots,s_m)=\sum_{n_1,n_2,\ldots,n_m\geq 0}\dfrac{1}{L_1^{s_1}L_2^{s_2}\cdots L_m^{s_m}}}}

where L_j^{s_j} are inhomogeneous functions of (n_1,n_2,\ldots,n_m). 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

\boxed{\displaystyle{\zeta_W (s)=\sum_{R}\dfrac{1}{\mbox{dim}(R)^s}}}

This sum is taken over the equivalence classes of irreducible representations R of G. Considering a root system \Delta of rank equal to r and with n positive roots in \Delta^+, being all simple without loss of generality, the simple roots \lambda_i allow us to define the Witten zeta function as a function of several variables:

\boxed{\displaystyle{\zeta_W (s_1,s_2,\ldots,s_n)=\sum_{m_1,m_2,\ldots,m_r> 0}\prod_{\alpha \in \Delta^+}\left[\dfrac{1}{\left(\alpha^V,m_1\lambda_1+m_2\lambda_2+\ldots+m_r\lambda_r\right)}\right]}}

Zeta function of an operator

The zeta function of any (pseudo)-differential operator \mathcal{P}, or more generally any operator, can be defined as the following functional series:

\boxed{\displaystyle{\zeta_{\mathcal{P}} (s)=\mbox{Tr}_\zeta (\mathcal{P}^{-s})}}

and where the trace \mbox{Tr}_\zeta 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:

\boxed{\displaystyle{\zeta_{\mathcal{P}} (s)=\sum_{\lambda_i}\lambda_i^{-s}}}

It allow us to define the generalization of the determinant to \infty-dimensional operators in the following non-trivial way:

\boxed{\displaystyle{\det_{\zeta} \mathcal{P}=e^{-\zeta_{\mathcal{P}}^{'}(0)}}}

Dirichlet L-function/L-series

They are the formal series

\boxed{\displaystyle{L(\chi,s)=\sum_n\dfrac{\chi (n)}{n^s}=\prod_{p\;\; prime}\dfrac{1}{1-\chi (p)p^{-s}}}}

where \chi is a Dirichlet character with conductor f, i.e.,

\displaystyle{\sum_ {n=0}^\infty B_{n,\chi}\dfrac{t^n}{n!}=\sum_{n=1}^f\dfrac{\chi (n)te^{nt}}{e^{ft}-1}}

There, the generalized Bernoulli numbers are related to the L-series through the generating function above, and they satisfy the identity

L(1-n,\chi)=-\dfrac{B_{n,\chi}}{n}

p-adic zeta function

The p-adic analogue of the zeta function is defined with the following equation:

\zeta_p (s)\equiv \dfrac{1}{1-p^{-s}}

Moreover, we also define the zeta function at the infinite real prime:

\zeta_\infty(s) \equiv \pi^{-s/2}\Gamma \left(\dfrac{s}{2}\right)

The p-adic zeta function and the “real” prime zeta function (zeta function in the so-called “infinite prime”) satisfy the important adelic identity:

\displaystyle{\zeta_\infty (s)\prod_{p=2}^\infty \zeta_p (s)=\zeta_{\mathbb{A}}(s)}

where \zeta_{\mathbb{A}} (s)=\zeta_\infty (s)\zeta (s),  and \zeta (s) is the classical Riemann zeta function. This adelic identity is just a special case of the adelic-type identity:

\displaystyle{\vert x\vert_\infty \prod_p\vert x\vert_p=1}

Stay tuned…The great adventure of Physmatics is just beginning!

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