Casimir Force, Casimir Operators and the Riemann Hypothesis, 1–7  © de Gruyter 2010

Absolute zeta functions, absolute Riemann hypothesis and absolute Casimir energies

Shin-ya Koyama and Nobushige Kurokawa

Abstract. We prove that the absolute zeta functions for finite abelian monoids have positive Casimir energy if and only if its order is divisible by 4.

Keywords. Absolute zeta functions, absolute mathematics, the field with one element, F_{1}, the Casimir energy.

AMS classification. 11M06.

1 Introduction

In 1948, Casimir used the value

ζ(−1) = −\frac{1}{12}  (1.1)

for the Riemann zeta function

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}}

for calculating a certain force between two metal objects; see Casimir [C], Hawking [Haw], and Kurokawa-Wakayama [KW1]; we remark that ζ(−1) appears in the one dimensional case and that ζ(−3) is used in the ordinary three dimensional situation. The value (1.1) represents the value of a divergent series

1 + 2 + 3 + · · · .

When a zeta function

Z(s) = ∑_{λ} λ^{−s}

is given, we call Z(−1) its Casimir energy, which is named after the work of Casimir.

From various viewpoints, the signature of the Casimir energy is meaningful in the sense that it reflects certain dynamical and arithmetical properties. For example, the following proposition is known.

Proposition 1.1. Let K be a finite extension of Q with its integer ring O_{K}. The Casimir energy for the Dedekind zeta function

ζ_{K}(s) = ∑_{{I⊂O_{K} \atop ideal≠0}} N(I)^{−s}

with N(I) = |O_{K}/I| is as follows:

ζ_{K}(−1) is \begin{cases}positive & ⟺ K is totally real and [K : Q] is even, \\ negative & ⟺ K is totally real and [K : Q] is odd, \\ 0 & ⟺ K is not totally real. \end{cases}

For example, ζ(−1) = −1/12 is actually negative for K = Q.

In this paper, we prove an analog of Proposition 1 for the absolute Weil zeta function. We recall briefly the history of absolute mathematics. The absolute mathematics has been studied in many papers after Tits[T] (1957) as in the papers listed in the references. It is recommended to read the excellent survey of Manin [M] published in 1995. Early publications are [K1](1992) and [KOW](2003); see also [CC], [D], [Har], [K2], [KO], [KW2], and [S]. The absolute zeta functions were studied recently in papers [DKK] and [KKK]. Here we describe the definitions:

Let μ μ μ_{N} = {z ∈ C | z^{N} = 1} (N = 1, 2, 3, ...) and put

F_{1^{N}} = μ μ μ_{N} ∪ {0}.

The absolute Weil zeta function is defined in [DKK] by

ζ_{F_{1^{N}}}(s) = exp(∑_{m=1}^{∞} \frac{|Hom(F_{1^{N}}, F_{1^{m}})|}{m}e^{−ms}) .

More generally, for a fintie abelian group G, we put F_{1}[G] = G ∪ {0}, which is a multiplicative monoid (an F_{1}-algebra). We define the absolute Weil zeta function of F_{1}[G] by

ζ_{F_{1}[G]}= exp(∑_{m=1}^{∞} \frac{|Hom(F_{1}[G], F_{1^{m}})|}{m}e^{−ms}) .

In particular, we have F_{1^{N}} = F_{1}[μ μ μ_{N}].

These absolute zeta functions are constructed for F_{1}-algebras, where F_{1} is “the field with one element". Precisely speaking we understand that “an F_{1}-algebra" means “a multiplicative monoid with 0". Especially F_{1} = {1, 0}.

These absolute zeta functions are considered as absolute version of the classical Weil (or congruence) zeta functions. In the classical case, we are looking at the number of rational points over finite fields. In the absolute case, we look at the number of rational points over finite extensions of F_{1}.

In general they satisfy analytic continuations, functional equations and the associated Riemann hypothesis. Special values at negative integers are considered to be absolute Casimir energies. Here we restrict to the zeta function of a finite extension of F_{1}.

For describing our earlier results, we introduce the absolute Frobenius operator Φ_{G} for a finite abelian group G, which operates on μ μ μ_{N^{2}} with |G| = N. First when G = μ μ μ_{N}, we define

Φ_{G} : μ μ μ_{N^{2}} ∋ α ⟼ α^{N+1} ∈ μ μ μ_{N^{2}}.

For general cases, put

G ≅ μ μ μ_{N_{1}} × ⋯ × μ μ μ_{N_{r}}

with N = N_{1} ⋯ N_{r}. Then

Φ_{G} = Φ_{N_{1},...,N_{r}} : μ μ μ_{N_{1}^{2}} × ⋯ × μ μ μ_{N_{r}^{2}} ⟶ μ μ μ_{N_{1}^{2}} × ⋯ × μ μ μ_{N_{r}^{2}}

is defined by the Kronecker tensor product

Φ_{G} = Φ_{N_{1}} ⊗ ⋯ ⊗ Φ_{N_{r}}.

In [DKK] and [KKK], we proved the following theorem:

Theorem 1.2. For any finite abelian group G, the following properties hold:

(i) ζ_{F_{1}[G]}(s) has analytic continuation to all the complex numbers.

(ii) ζ_{F_{1}[G]}(s) satisfies the associated functional equation

ζ_{F_{1}[G]}(−s) = w(G)e^{−|G|s}ζ_{F_{1}[G]}(s),

where w(G) is a complex number of modulus 1.

(iii) ζ_{F_{1}[G]}(s) satisfies the analogue of the Riemann hypothesis. Namely, all singularities of ζ_{F_{1}[G]}(s) are on the line Re(s) = 0.

(iv) ζ_{F_{1}[G]}(s) has the determinant expression

ζ_{F_{1}[G]}(s) = det(1 − Φ_{G}e^{−s}) ^{−1/|G|} .  (1.2)

We define the normalized Casimir energy for the absolute zeta function of an F_{1}- algebra A by

ζ_{A}(−1)^{|End(A)|}.

Our main result is as follows:

Theorem 1.3. Let A be a fintie abelian F_{1}-algebra. Then we have

ζ_{A}(−1)^{|End(A)|} > 0 ⟺ 4 | [A : F_{1}].

2 Proofs

We start by proving a special case where A = F_{1^{N}}. The following Lemma will be useful in the proof.

Lemma 2.1. The integer

a(N) = ∑_{n|N} \frac{N}{n} φ(n)

is even, if and only if 4|N.

Proof. The sequence a(N) is a convolution of b(N) = N and c(N) = φ(N), since

(b ∗ c)(N) = ∑_{mn=N} b(m)c(n) = ∑_{n|N} b (\frac{N}{n} ) c(n) = ∑_{n|N} \frac{N}{n} φ(n) = a(N).

Because both b(N) and c(N) are multiplicative, the convolution a(N) is also multiplicative. Therefore

a(N) = ∏_{{p|N \atop p: prime}} a(p^{ord_{p}(N)}).

Hence a(N) is odd if and only if a(p^{ord_{p}(N)}) is odd for all p|N. By definition it holds that a(1) = 1 and

a(p^{l}) = ∑_{k=0}^{l}p^{l−k}φ(p^{k}).  (2.1)

We will examine the condition for (2.1) being odd. When p is odd, we compute

a(p^{l}) = p^{l}φ(1) + ∑_{k=1}^{l}p^{l−k}φ(p^{k})

= p^{l} + ∑_{k=1}^{l}p^{l−k}p^{k−1}(p − 1)

= p^{l} + (p − 1)p^{l−1}l,

which is always odd. When p = 2, we compute

a(2^{l}) = 2^{l} + 2^{l−1}l

= \begin{cases}odd & (l = 1) \\ even & (l ≥ 2). \end{cases}

Consequently, (2.1) is odd if and only if either p is odd or (l, p) = (1, 2). In other words, (2.1) is even if and only if ord_{2}(N) ≥ 2, which equivalently is 4|N.

Proposition 2.2.

ζ_{F_{1^{N}}}(−1)^{N} > 0 ⟺ 4 | N.

Proof. By [DKK] we have an Euler product expression

ζ_{F_{1^{N}}}(s)^{N} = ∏_{n|N}(1 − e^{−ns})^{− \frac{N}{n} φ(n)}.

By this expression we have

sgn(ζ_{F_{1^{N}}}(−1)^{N}) = (−1)^{∑_{n|N} \frac{N}{n} φ(n)}

= \begin{cases}1 & if ∑_{n|N} \frac{N}{n} φ(n) is even, \\ −1 & if ∑_{n|N} \frac{N}{n} φ(n) is odd. \end{cases}

Then by the previous lemma, it holds that

sgn(ζ_{F_{1^{N}}}(−1)^{N}) = \begin{cases}1 & if 4|N, \\ −1 & if 4 |̸N. \end{cases}

Proof of Theorem 1:

By the determinant expression (1.2) proved in [KKK], we have

ζ_{F_{1^{N}}}(−1)^{N} = det(1 − Φ_{N}e)^{−1}

= det((−Φ_{N}e)(Φ_{N}^{−1}e^{−1} − 1))^{−1}

= (−1)^{N^{2}} det(Φ_{N})e^{−N^{2}} det(1 − Φ_{N}e^{−1})^{−1}

= (−1)^{N^{2}} det(Φ_{N})e^{−N^{2}}ζ_{F_{1^{N}}}(1)^{N}

= (−1)^{N^{2}} det(Φ_{N})e^{−N^{2}} ∏_{n|N}(1 − e^{−n})^{− \frac{N}{n} φ(n)}.

Hence

sgn(ζ_{F_{1^{N}}}(−1)^{N}) = (−1)^{N^{2}} det(Φ_{N})
= (−1)^{N^{2}}sgn(Φ_{N}).

By the preceding proposition, this is negative if and only if 4 |̸N. Thus

sgn(Φ_{N}) = \begin{cases}1 & if N ≡ 1, 3 (mod 4), \\ −1 & if N ≡ 2 (mod 4). \end{cases}  (2.2)

6  S. Koyama and N. Kurokawa

By [KKK] we have the determinant expression

ζ_{F_{1}[G]}(s)^{N} = det(1 − Φ_{G}e^{−s})^{−1}.

Thus we compute

sgn(ζ_{F_{1}[G]}(−1)^{N}) = (−1)^{N^{2}} det(Φ_{N_{1},...,N_{r}})
= (−1)^{N^{2}}sgn(Φ_{N_{1}})^{N_{2}^{2}···N_{r}^{2}} ⋯ sgn(Φ_{N_{r}})^{N_{1}^{2}···N_{r−1}^{2}}.

First when all of N_{1}, ..., N_{r} are odd, we find by (2.2) that sgn(ζ_{F_{1}[G]}(−1)^{N}) = −1.

Next when 2|N_{1} ⋯ N_{r} but 4 |̸N_{1} ⋯ N_{r}, which means only one of N_{j} is 2 (mod 4) and all others are odd, we see again from (2.2) that sgn(ζ_{F_{1}[G]}(−1)^{N}) = −1.

Finally, when 4|N_{1} ⋯ N_{r}, it holds that sgn(ζ_{F_{1}[G]}(−1)^{N}) = 1.

This completes the proof of Theorem.

Bibliography

[C] H.B.G. Casimir: On the attraction between two perfectly conducting plates, Proc. Kon. Nederland. Akad. Wetensch. B51 (1948), 793-795.

[CC] A. Connes and C. Consani: Schemes over F_{1} and zeta functions, arXiv 0903.2024v3 [math. AG] 9 Jul 2009.

[D] A. Deitmar: Schemes over F_{1}, Progress in Math. 239 (2005), 87-100.

[DKK] A. Deitmar, S. Koyama and N. Kurokawa: Absolute zeta functions, Proc. Japan Acad. 84A (2008), 138-142.

[Har] S. Haran: Non-additive geometry, Compos. Math. 143 (2007), 618-688.

[Haw] S.W. Hawking: Zeta function regularization of path integrals in curved space time, Comm. Math. Phys. 55 (1977), 133-148.

[K1] N. Kurokawa: Multiple zeta functions: an example, Zeta Functions in Geometry (Tokyo 1990), 219-226, Adv. Stud. in Pure Math. 21, Kinokuniya, Tokyo, 1992.

[K2] N. Kurokawa: Zeta functions over F_{1}, Proc. Japan Acad. 81A (2005), 180-184.

[KKK] S. Kim, S. Koyama and N. Kurokawa: The Riemann hypothesis and functional equations for zeta functions over F_{1}, Proc. Japan Acad. 85A (2009), 75-80.

[KO] N.Kurokawa and H. Ochiai: A multivariable Euler products of Igusa type and its applications, J. Number Theory 129 (2009), 1919-1930.

[KOW] N.Kurokawa, H. Ochiai and M. Wakayama: Absolute derivations and zeta functions, Doc. Math. 2003, 565-584.

[KW1] N.Kurokawa and M. Wakayama: Casimir effects on Riemann surfaces, Indag. Math. (N.S.) 13 (2002), 63-75.

[KW2] N.Kurokawa and M. Wakayama: Absolute tensor products, International Math. Research Notices 2004, no. 5, 249-260.

Absolute zeta functions  7

[M] Yu. I. Manin: Lectures on zeta functions and motives (according to Deninger and Kurokawa), Columbia University Number Theory Seminar (New York, 1992), Asterisque 228 (1995), 121-163.

[S] C. Soulé: Les variete sur le corps á un element, Moscow Math. J. 4 (2004) , 217-244.

[T] J. Tits: Sur les analogues algebriques des groupes semi-simple complexes, Colloque d’algebre superieure, tenu a Bruxelles du 19 au 22 decembre 1956, Centre Belge de Recherches Math., Gauthier-Villar,Paris 1957, 261-289.

Author information

Shin-ya Koyama, Department of Biomedical Engineering, Toyo University, 2100 Kujirai, Kawagoe-shi 350-8585, Japan.

E-mail: koyama@toyonet.toyo.ac.jp

Nobushige Kurokawa, Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan.

E-mail: kurokawa@math.titech.ac.jp