Search

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]

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]

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]

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]

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.

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]

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]

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]

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]

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]

We study lower bounds of a general family of L-functions on the 1-line. More precisely, we show that for any F(s) in this family, there exist arbitrarily large t such that F(1+it) ≥ eγF (log2t + log3t)m + O(1), where m is the order of the pole of F(s) at s = 1. This is a generalisation of the result of Aistleitner, Munsch and Mahatab ['Extreme values of the Riemann zeta function on the 1-line', Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type L(s,ƒ × ƒ) on the 1-line. [ABSTRACT FROM AUTHOR]

In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï's formula concerning the error term in the Dirichlet divisor problem. [ABSTRACT FROM AUTHOR]

Let K be a number field of degree k ≥ 8. In this article, we investigate a certain density theorem for zeros of the Dedekind zeta function attached to K. In this context, our theorem strengthens a result of Heath-Brown. Further, we investigate an upper-bound of the error term for an ideal counting function attached to K. When K is a cyclotomic field, we prove a stronger upper-bound for this error term. [ABSTRACT FROM AUTHOR]

Let K3 be a non-normal cubic extension over ℚ:We study the higher moment of the coefficients of Dedekind zeta function over sum of two squares where 2 ⩽ l ⩽ 8 and n1; n2; l∈ ℤ. [ABSTRACT FROM AUTHOR]

This paper is a corrigendum to the article "On the ideal theorem for number fields". The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term. [ABSTRACT FROM AUTHOR]

Abstract We consider a certain class of multiplicative functions f : N → C. Let F (s) = ∑ n = 1 ∞ f (n) n − s be the associated Dirichlet series and F N (s) = ∑ n ≤ N f (n) n − s be the truncated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of f. More precisely, we prove estimates for the sum ∑ n = 1 x f (n) / n in terms of the size of | F (1 + 1 / log ⁡ x) | and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for the partial sums F N (s). In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field K. More precisely, we give some improved results for the number of zeros up to height T as well as new zero density results for the number of zeros up to height T , lying to the right of Re (s) = σ , where σ ≥ 1 / 2. [ABSTRACT FROM AUTHOR]

The Stieltjes constants γn(K) of a number field K are the coefficients of the Laurent expansion of the Dedekind zeta function ςK(s) at its pole s = 1. In this paper, we establish a similar expression of γn(K) as Stieltjes obtained in 1885 for γn(Q). We also study the signs of γn(K). [ABSTRACT FROM AUTHOR]

For a positive integer N, we derive a Kronecker type limit formula for a Dedekind type zeta function ζ O K (N) (s ; [ a ]) associated to a wide ideal class of a quadratic order of conductor N of a quadratic field K. As consequences, we establish a Chowla–Selberg type formula for a modular form for Γ 0 (N) , give a proof to a classical class number relation of a quadratic field, and find an asymptotic formula for the average value of the number of representations by a primitive positive definite quadratic form of discriminant N 2 d with d fundamental. Furthermore, as an application, we obtain a formula for the Petersson norm of a theta function attached to a class character of the ideal group of conductor N of O K . [ABSTRACT FROM AUTHOR]

We give an extension of a theorem of Cesàro from the rational integers to the ring of integers of an arbitrary number field. This extension is used to generalize Pillai's function to number fields. [ABSTRACT FROM AUTHOR]

We consider the number of visible lattice points under the assumption of the Extended Lindelöf Hypothesis. We get a relation between visible lattice points and the Extended Lindelöf Hypothesis. And we also get a relation between visible lattice points over Q ( − 1 ) and the Gauss Circle Problem. [ABSTRACT FROM AUTHOR]

Let χ range over the non-trivial primitive characters associated with the abelian extensions L / K of a given number field K , i.e. over the non-trivial primitive characters on ray class groups of K . Let f χ be the norm of the finite part of the conductor of such a character. It is known that | L ( 1 , χ ) | ≤ 1 2 Res s = 1 ( ζ K ( s ) ) log ⁡ f χ + O ( 1 ) , where the implied constants in this O ( 1 ) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of χ , i.e. ramified prime ideals. This has already been done only in the case that K = Q . This paper is devoted to giving for the first time such improvements for any K . As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field. [ABSTRACT FROM AUTHOR]

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]