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We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit lower bound for the rank of universal quadratic forms over a given number field K, under the assumption that the codifferent of K is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields. [ABSTRACT FROM AUTHOR]

Two algebraic number fields are called arithmetically equivalent if the Dedekind zeta functions of the fields coincide. We show that if a G-extension contains non-conjugate arithmetically equivalent fields and there is an injection from G to another group H inducing an isoclinism between G and H, then there are non-conjugate arithmetically equivalent fields inside an H-extension. [ABSTRACT FROM AUTHOR]

The Northcott property for special values of Dedekind zeta functions and more general motivic L-functions was defined by Pazuki and Pengo. We investigate this property for any complex evaluation of Dedekind zeta functions. The results are more delicate and subtle than what was proven for the function field case in previous work of Li and the authors, since they include some surprising behavior in the neighborhood of the trivial zeros. The techniques include a mixture of analytic and computer-assisted arguments. [ABSTRACT FROM AUTHOR]

Given a number field, the Euler–Kronecker constant is defined as the constant term in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function at the point s = 1 . In the case of real and imaginary quadratic fields, a closed-form expression for the Euler–Kronecker constants can be obtained with the help of suitable Kronecker limit formulas. In this article, we avoid the use of Kronecker limit formulas and derive an explicit series representation of these constants with the help of an asymptotic series representation of the coefficients appearing in the Laurent series expansion of the Dedekind zeta function at s = 1 . As a result, the expressions obtained do not require evaluation of the special functions appearing in the Kronecker limit formulas. [ABSTRACT FROM AUTHOR]

Let K3 be a non-normal cubic extension over Q. In this paper, we investigate the higher moments of coefficients aK3 (n) of the Dedekind zeta function of the following type Σ(n≤x) al K3 (n), where l ≥ 4 is any fixed positive integer. [ABSTRACT FROM AUTHOR]

In this article, using the arithmetic of Hurwitz zeta functions and Gauss sums, we investigate the arithmetic nature of ζ K (2 k + 1) ζ K + (2 k + 1) , where K is any totally imaginary abelian extension with maximal totally real subfield K + and k ≥ 1 is an integer. As a consequence, we derive an elementary proof of the elegant result of Murty and Pathak about irrationality of Dedekind zeta values associated to imaginary quadratic fields at odd integers. This alternate line of action allows us to link these questions to a question of Chowla and Milnor, resulting in some new results about Dedekind zeta values. [ABSTRACT FROM AUTHOR]

Let F/k be a finite abelian extension of number fields with k imaginary quadratic. Let OF be the ring of integers of F and n ⩾ 2 a rational integer. We construct a submodule in the higher odd-degree algebraic K-groups of OF using corresponding Gross's special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher "twisted" class number of F, which is the cardinal of the finite algebraic K-group K2n−2(OF). [ABSTRACT FROM AUTHOR]

For a number field K , and integral ideals I and J in its number ring O K , Nowak studied the asymptotic behavior of the average of Ramanujan sums C J (I) over both ideals I and J . In this article, we extend this investigation by establishing asymptotic formulas for the second moment of averages of Ramanujan sums over quadratic and cubic number fields, thereby generalizing previous works of Chen, Kumchev, Robles, and Roy on moments of averages of Ramanujan sums over rationals. Additionally, using a special property of certain integral domains, we obtain second moment results for Ramanujan sums over some other number fields. [ABSTRACT FROM AUTHOR]

We study the class number one and class number two problems for real quadratic fields Q (9 m 2 + 4 m) , where m is an odd integer. We show that for m ≡ 1 (mod 3) there is only one such field with class number one and only one such field with class number two. Moreover, we also evaluate some generalized Dedekind sums. [ABSTRACT FROM AUTHOR]

Denote by $ a_{K}(n) $ the number of integral ideals in $ K $ with norm $ n $, where $ K $ is a algebraic number field of degree $ m $ over the rational field $ \mathcal{Q} $. Let $ p $ be a prime number. In this paper, we prove that, for two distinct quadratic number fields $ K_i = \mathcal{Q}(\sqrt{d_i}), \ i = 1, 2 $, the sets both $ \{p\ |\ a_{K_1}(p)< a_{K_2}(p)\} \text{ and } \{p\ |\ a_{K_1}(p^2)< a_{K_2}(p^2)\} $ have analytic density $ 1/4 $, respectively.

Under GRH, we establish a version of Duke's short-sum theorem for entire Artin L -functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit. [ABSTRACT FROM AUTHOR]

Let K be a number field over Q and let a K (m) denote the number of integral ideals of K of norm equal to m ∈ N. In this paper we obtain asymptotic formulae for sums of the form ∑ m ≤ X a K l (m) thereby generalizing the previous works on the problem. Previously such asymptotics were known only in the case when K is Galois or when K was a non normal cubic extension and l = 2 , 3. The present work subsumes both these cases. [ABSTRACT FROM AUTHOR]

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form Δ should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion. [ABSTRACT FROM AUTHOR]

For a number field K , let ζ K (s) be the Dedekind zeta function associated to K. In this paper, we study non-vanishing and transcendence of ζ K as well as its derivative ζ K ′ at s = 1 / 2. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of e γ and non-vanishing of L -series at s = 1 / 2 , J. Number Theory 161 (2016) 444–456]. [ABSTRACT FROM AUTHOR]

We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for x ≥ 2 . Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at s = 1 . [ABSTRACT FROM AUTHOR]

For any number field, we define Dedekind harmonic numbers with respect to this number field. First, we show that they are not integers except finitely many of them. Then, we present a uniform and an explicit version of this result for quadratic number fields. Moreover, by assuming the Riemann hypothesis for Dedekind zeta functions, we prove that the difference of two Dedekind harmonic numbers are not integers after a while if we have enough terms, and we prove the non-integrality of Dedekind harmonic numbers for quadratic number fields in another uniform way together with an asymptotic result. [ABSTRACT FROM AUTHOR]