Factorization and independence of arithmetic functions

In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π 2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζ G (2) where ζ G is the zeta function associated with the Gaussian integers. In order to calculate the Gaussian probability, we use a theorem that enables us to factor the zeta function into a product of the Riemann zeta function and a Dirichlet series called an L-series. For the Gaussian integers we get: ζ G (2)=ζ(2)L(2,χ), where $$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$ We use this factorization to approximate the Gaussian probability to 17 decimal places.

We investigate the problem on the validity of the Lazard theorem analogue (or the Kronecker–Vahlen theorem), which states that the least remainder Euclidean Algorithm (EA) has the shortest length over any other versions of EA, in unique factorization domains. There is obtained the existence criterion to represent a fixed element of the fractions field of a unique factorization domain in the form of a continued fraction of a fixed length. This criterion enables to obtain a formula for the length of the least remainder (on norm) EA as a function of elements, with respect to which EA is applied. This result gives us the class T of unique factorization domains, for which the Lazard theorem analogue is valid. We propose algorithms to check whether the given unique factorization domain belongs to the class T. We find the necessary and sufficient conditions under which the number of steps in the worst case of the least remainder EA grows not faster than logarithm. In particular, these results hold for the least remainder EA in any Euclidean quadratic domain. We provide counterexamples, which show the essentiality of the conditions in the obtained theorems.

Some problems of the number theory are associated with the zeros of special functions, such as the Riemann zeta function ζ(s), Dirichlet L-functions L(s, χ) and others. The Riemann zeta function is the most famous. On the half-plane ℜs > 1, the Riemann zeta function is defined by Dirichlet series ζ(s) = ∑ +∞ n=1 n −s . In 1859, Riemann conjectured that non-trivial zeros of the Riemann zeta function lie on the critical line ℜs = 1 2 . Hardy was the first to prove in 1914 that an infinity of zeros are on the critical line. In 1942, Selberg obtained lower bound of the correct order of magnitude for the number zeros of the Riemann zeta functions on intervals of critical line [T, T + H], H = T 0.5+e . In 1984, A. A. Karatsuba proved Selberg’s result for shorter intervals of critical line [T, T + H], H = T 27 82 +e . For arithmetic Dirichlet series satisfying a functional equation of Riemann type but admitting no Euler product expansions, lower bounds of the correct order of magnitude for the number of their zeros on intervals of the critical line ℜs = 1/2 have not been obtained so far. The first to show that the critical line contains abnormally many zeros of an arithmetic Dirichlet series without Euler products was Voronin, who proved in 1980 that interval (0, T] of critical line contains more than cT e 1 20 √ ln ln ln ln T zeros of the Davenport–Heilbronn function. In 1989 A. A. Karatsuba developed a new method for obtaining lower bounds for the number zeros of certain Dirichlet series in intervals of critical line; by using this method, he substantially strengthened Voronin’s result. In 1991 Karatsuba solved (by his 1989 method) the problem of estimating the number zeros of linear combinations of functions which are analogous the Hardy function. In the present paper we prove a theorem similar to the theorem of A. A. Karatsuba (in 1991), but only for almost all intervals of the form (T, T + H), H = Xe , where e is an arbitrary positive number, and X ≤ T ≤ 2X, X > X0(e).

For a lattice in two dimensions, the number of distinct derivative lattices of index n is given by the arithmetic function σ1(n) which is the sum of the divisors of n, including 1 and n. The function σ1(n) has as its generating function the Dirichlet series ζ(s)ζ(s −1) where ζ(s) = Σ∞n=1n−s is the Riemann zeta function. That is, ζ(s)ζ(s −1) = Σ∞n = 1 σ1(n)n−s. The probability that s points chosen at random on the two-dimensional lattice do not lie on any of the derivative lattices so enumerated is therefore [ζ(s)ζ(s − 1)]−1. The equivalent results in three dimensions are: the arithmetic function Σd|n[n/d]2σ1(d), where the sum is over the divisors d of n, the generating function ζ(s)ζ(s − 1)ζ(s − 2) and the probability [ζ(s)ζ(s − 1)ζ(s − 2) ] − 1. Applied to the reciprocal lattice, this provides a method of estimating whether such a particular non-primitive arrangement of strong reflections could occur by chance. This number-theory approach may be adopted to the enumeration of derivative lattices in the general case. However, when considering potential sublattices in practice, only those belonging to the same Laue class are of any interest, in which case the general formula only holds for the Laue class \overline 1. For all other space groups, the effect must be considered of choosing s points at random together with all the other points related to them through the diffraction symmetry. This leads to a generating function that is identical for space groups belonging to the same Patterson symmetry, that is Laue class and lattice type. In all 24 cases, the form is A(s)/F(s) where F depends only on the Laue class and is a product of infinite series, chiefly zeta functions, but also Dirichlet L functions. A(s) in turn derives from the lattice type, but varies depending on what other lattice types are available as potential sublattices in that Laue class. It represents an adjustment to one prime-number term in the infinite-product form of F, it being the p = 2 term in the monoclinic, orthorhombic, tetragonal and cubic crystal classes and the p = 3 term in the trigonal and hexagonal classes. The numerous results concerning generating functions, arithmetic functions and probabilities are given in the tables.

This chapter treats a class of series, called general Dirichlet series, which includes both power series and ordinary Dirichlet series as special cases. Most of the chapter is devoted to a method developed by Harald Bohr [6] in 1919 for studying the set of values taken by Dirichlet series in a half-plane. Bohr introduced an equivalence relation among Dirichlet series and showed that equivalent Dirichlet series take the same set of values in certain half-planes. The theory uses Kronecker’s approximation theorem discussed in the previous chapter. At the end of the chapter applications are given to the Riemann zeta function and to Dirichlet L-functions.

Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring consisting of all power series in such that for all large i. By definition, the polynomial ring and the power series ring are special cases of when and I = D, respectively. In this article, we investigate the ring in the case I is a nonzero proper ideal of D. We prove that R is a two-dimensional non-Noetherian ring. For each maximal ideal P of D, it is shown that is a height-one prime ideal of R. The set of units of R is given and the spectrum of R is also described. Unlike the power series ring the ring R is not a unique factorization domain (UFD). Furthermore, when I is a nonzero prime ideal, R does not satisfy both ACCP and the atomic property. In obtaining results on R, we introduce and sometimes use results on the ring RS , where I = dD with and Closely related to R, the ring RS is shown to be a Noetherian UFD with Krull dimension at most two. Moreover, RS has Krull dimension two exactly when I is not contained in the Jacobson radical of D; otherwise RS is a PID.

Let q be a power of a prime p and Fq be the finite field with q elements. Denote by Fq[X] and Fq(X) the ring of polynomials with coefficients in Fq and the quotient field of Fq[X] respectively. For each P/Q ∈ Fq(X) define |P/Q| = qdeg(P )−deg(Q) where for an element g ∈ Fp[X]we have denoted its degree by deg(g). Note that with respect to the valuation function | · | the integral domain Fq[X] is a Euclidean domain and so also a principal ideal domain and a unique factorization domain. In particular Bezout’s identity is satisfied. Let Fq((X−1)) denote the field of formal Laurent series Fq((X)) = {anX + · · · + a0 + a−1X + · · · : n ∈ Z, ai ∈ Fq},

1: Rings. 2: Euclidean rings. 3: Highest common factor. 4: The four-squares theorem. 5: Fields and polynomials. 6: Unique factorization domains. 7: Field of quotients of an integral domain. 8: Factorization of polynomials. 9: Fields and field extensions. 10: Finite cyclic groups and finite fields. 11: Algebraic numbers. 12: Ruler and Compass constructions. 13: Homomorphisms, ideals and factor rings. 14: Principal ideal domains and a method for constructing fields. 15: Finite fields. Solutions to selected exercises. References

Five basic mistakes are found in the Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane and in the domain of the function, when the left-hand side of equation is finite, the right-hand side may be infinite, and vice versa. The Riemann Zeta function equation holds only at the point Re(s) = 1/2(s= a+ \(i b\)) . However, at this point, the Zeta function is infinite, rather than zero, to make Riemann hypothesis untenable. 2. An integral item around the original point of coordinate system was ignored when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1but divergent when Re(s) < 1 .The integral form of Zeta function does not change the divergence of its series form. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x > 0 . At point x = 0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x = 0 , so the formula can not be used. 4. Because the integral lower limit of Zeta function is zero, the integrand function is not uniformly convergent, so integral sign and sum sign can not be exchanged. But Riemann made them interchangeable, resulting in that the integral form of Zeta function is untenable. 5.The formula \(\theta(x)=\sqrt{x} \theta(1 / x)\) of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also x > 0 . Because the lower limit of integral in the deduction was x = 0 , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, the analytic property of the original function is destroyed and the Cauchy-Riemann equation can not be satisfied. So they are not the real zeros of strict Riemann Zeta function.

Louis Solomon introduced the notion of a zeta function ${\zeta _\Theta }(s)$ of an order $\Theta$ in a finite-dimensional central simple K-algebra A, with K a number field or its completion ${K_P}$ (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate ${\zeta _\Theta }(s)$, it is sufficient to consider the zeta function of local orders ${\Theta _P}$. However, since these local orders ${\Theta _P}$ are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’, ${Z_{{\Theta _P}}}(s)$, which is related to the free ${\Theta _P}$-ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., $[A:K] = 4$). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function ${Z_{{\Theta _P}}}(s)$ of any local hereditary order ${\Theta _P}$ in a central simple algebra over a local field. We obtain ${Z_{{\Theta _P}}}(s)$ as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.

Euclidean domain is also known as Euclidean ring. It is an integral domain which is associated with at least one Euclidean function [1]. A non zero commutative ring in which of product two non zero element is non zero is known as integral domain. Integral domain is generalization of ring of integers [2]. In Euclidean domain we use Euclidean algorithm for calculating greatest common divisors of two elements. Greatest common divisors of two elements always exist and this can be represented in linear combination of these elements. Euclidean domain has ideals and every ideal of Euclidean domain is principal ideal [3]. Euclidean domain has property of unique factorization domain and every Euclidean domain is a unique factorization domain it means every non zero non unit element can be written as product of prime elements. Key words: algorithm, ideal, trichotomy, polynomials

In this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

Symmetries of Bernoulli polynomial series and Arakawa–Kaneko zeta functions

Bounds for determinants of matrices associated with classes of arithmetical functions

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummers congruences play important roles in the investigation of p -adic interpolation of the classical Riemann zeta function.