Journal of Number Theory

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Li's criterion for Epstein zeta functions, generalization of Kronecker's limit formula and the Gauss problem

Francis Brown, Sami Omar

Journal of Number Theory > 2016 > 158 > C > 90-103

In this paper, we first investigate the Li criterion for the Riemann hypothesis for a class of Epstein zeta functions. Then, we derive some interesting consequences for the class number of a quadratic imaginary number field K using Kronecker's limit formula and Hecke's formulation for the meromorphic continuation of the Dedekind zeta function ζK.

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On symmetries of elliptic nets and valuations of net polynomials

Amir Akbary, Jeff Bleaney, Soroosh Yazdani

Journal of Number Theory > 2016 > 158 > C > 185-216

We give a generalization of a theorem of Ayad on valuations of division polynomials in the context of net polynomials. We also present a generalization of a theorem of Ward on symmetry of elliptic divisibility sequences to the case of elliptic nets.

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Pair correlation of fractional parts derived from rational valued sequences, II

Sneha Chaubey, Melinda Lanius, Alexandru Zaharescu

Journal of Number Theory > 2016 > 158 > C > 151-164

A rational valued vector sequence x→, for some fixed k and r∈N, is a map x→:Nk→Qr. In the present paper, we complement the results of [3] with a discussion on rational valued vector sequences. We investigate the pair correlation function for the fractional parts of sequences t→⋅x→, where x→ is a rational valued vector sequence and t→∈Rr. We offer a new class of sequences x→ whose pair correlation...

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The (S,{2})-Iwasawa theory

Su Hu, Min-Soo Kim

Journal of Number Theory > 2016 > 158 > C > 73-89

Iwasawa made the fundamental discovery that there is a close connection between the ideal class groups of Zp-extensions of cyclotomic fields and the p-adic analogue of Riemann's zeta functionsζ(s)=∑n=1∞1ns. In this paper, we show that there may also exist a parallel Iwasawa's theory corresponding to the p-adic analogue of Euler's deformation of zeta functionsϕ(s)=∑n=1∞(−1)n−1ns.

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The valuative capacity of subshifts of finite type

Youssef Fares, Samuel Petite

Journal of Number Theory > 2016 > 158 > C > 165-184

The characteristic sequence of a subset E of the integers enables to generalize the factorial. The asymptotic limit of this sequence, called the valuative capacity, is actually related to the transfinite diameter of this set embedded into the set of p-adic integers. We use tools of potential theory to obtain formulas to compute this capacity. As an example, we provide a family of sets, invariant by...

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Monogenity of totally real algebraic extension fields over a cyclotomic field

Nadia Khan, Shin-ichi Katayama, Toru Nakahara, Tsuyoshi Uehara

Journal of Number Theory > 2016 > 158 > C > 348-355

Let K be a composite field of a cyclotomic field kn of odd conductor n≧3 or even one ≧8 with 4|n and a totally real algebraic extension field F over the rationals Q and both fields kn and F are linearly disjoint over Q to each other. Then the purpose of this paper is to prove that such a relatively totally real extension field K over a cyclotomic field kn has no power integral basis. Each of the composite...

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Rational products of singular moduli

Yuri Bilu, Florian Luca, Amalia Pizarro-Madariaga

Journal of Number Theory > 2016 > 158 > C > 397-410

We show that with “obvious” exceptions the product of two singular moduli cannot be a non-zero rational number. This gives a totally explicit version of André's 1998 theorem on special points for the hyperbolas x1x2=A, where A∈Q.

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On the p-rank of tame kernel of number fields

Chaochao Sun, Kejian Xu

Journal of Number Theory > 2016 > 158 > C > 244-267

In this paper, the relations between p-ranks of the tame kernel and the ideal class group for a general number field are investigated. As a result, nearly all of Browkin's results about quadratic fields are generalized to those for general number fields. In particular, a p-rank formula between the tame kernel and the ideal class group for a totally real number field of odd degree is obtained when...

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Resultants of minimal polynomials of maximal real cyclotomic extensions

K. Alan Loper, Nicholas J. Werner

Journal of Number Theory > 2016 > 158 > C > 298-315

Define the nth real cyclotomic polynomial to be the minimal polynomial over Z of ζn+ζn−1, where ζn=e2πi/n is a primitive nth root of unity. We prove that the real cyclotomic polynomials can be formed from compositions of polynomials closely related to the Chebyshev polynomials of the first kind. We use these relations to determine the resultant of two real cyclotomic polynomials.

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Partition identities with mixed mock modular forms

George E. Andrews, Stephen Hill

Journal of Number Theory > 2016 > 158 > C > 356-364

There are countless partition identities of the type introduced by Basil Gordon. All the previously known ones were related to infinite products usually modular forms. In this paper we identify a further class of partition functions of the Basil Gordon genre that are in fact eta-products times Hecke-type series.

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Polynomial identities on eigenforms

Joseph Richey, Noah Shutty

Journal of Number Theory > 2016 > 158 > C > 333-347

In this paper, we fix a polynomial with complex coefficients and determine the eigenforms for SL2(Z) which can be expressed as the fixed polynomial evaluated at other eigenforms. In particular, we show that when one excludes trivial cases, only finitely many such identities hold for a fixed polynomial.

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Explicit estimates for the number of rational points of singular complete intersections over a finite field

Guillermo Matera, Mariana Pérez, Melina Privitelli

Journal of Number Theory > 2016 > 158 > C > 54-72

Let V⊂Pn(F‾q) be a complete intersection defined over a finite field Fq of dimension r and singular locus of dimension at most 0≤s≤r−2. We obtain an explicit version of the Hooley–Katz estimate ||V(Fq)|−pr|=O(q(r+s+1)/2), where |V(Fq)| denotes the number of Fq-rational points of V and pr:=|Pr(Fq)|. Our estimate improves all the previous estimates in several important cases. Our approach relies on...

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On a problem of countable expansions

Yuru Zou, Derong Kong

Journal of Number Theory > 2016 > 158 > C > 134-150

For a real number q∈(1,2) and x∈[0,1/(q−1)], the infinite sequence (di) is called a q-expansion of x ifx=∑i=1∞diqi,di∈{0,1}for all i≥1. For m=1,2,⋯ or ℵ0 we denote by Bm the set of q∈(1,2) such that there exists x∈[0,1/(q−1)] having exactly m different q-expansions. It was shown by Sidorov [18] that q2:=min⁡B2≈1.71064, and later asked by Baker [1] whether q2∈Bℵ0? In this paper we provide a negative...

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Cooper and Lam's conjecture for generalized Bell ternary quadratic forms

Werner Hürlimann

Journal of Number Theory > 2016 > 158 > C > 23-32

Bell's theorem determines the counting function of the ternary quadratic forms x2+by2+cz2, with b,c∈{1,2,4,8}, in terms of the number r3(n) of representations of n as a sum of three squares. Based on it we verify Cooper and Lam's conjecture for them. This result includes two new cases so far left open. Additionally, we show that the forms (b,c)=(2,16) and (b,c)=(8,16) are generalized Bell forms in...

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Spectral correspondences for Maass waveforms on quaternion groups

Terrence Richard Blackman, Stefan Lemurell

Journal of Number Theory > 2016 > 158 > C > 1-22

We prove that in most cases the Jacquet–Langlands correspondence between newforms for Hecke congruence groups and newforms for quaternion orders is a bijection. Our proof covers almost all cases where the Hecke congruence group is of cocompact type, i.e. when a bijection is possible. The proof uses the Selberg trace formula.

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Generalization of Hensel's lemma: Finding the roots of p-adic Lipschitz functions

Ekaterina Yurova Axelsson, Andrei Khrennikov

Journal of Number Theory > 2016 > 158 > C > 217-233

In this paper we consider the problem of finding the roots of p-adic functions. In the case, where the function is defined by a polynomial with integer p-adic coefficients, using Hensel's lifting lemma helps us find the roots of the p-adic function.We generalize Hensel's lifting lemma for a wider class of p-adic functions, namely, the functions which satisfy the Lipschitz condition with constant pα,α≥0,...

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A refinement of the Dress–Scharlau theorem

Se Wook Jang, Byeong Moon Kim

Journal of Number Theory > 2016 > 158 > C > 234-243

In 1982, Dress and Scharlau [1] found an upper bound for the norm of totally positive, additively indecomposable algebraic integers in real quadratic fields and showed that this bound is sharp if the norm of the fundamental unit is −1. In this paper, we prove that their upper bound is not extremal if the norm of the fundamental unit is +1 and establish a new upper bound that is sharp for a large family...

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Some new modular relations for the Rogers–Ramanujan type functions of order eleven with applications to partitions

Chandrashekar Adiga, N.A.S. Bulkhali, D. Ranganatha, H.M. Srivastava

Journal of Number Theory > 2016 > 158 > C > 281-297

In this paper, we establish several modular relations for the Rogers–Ramanujan type functions of order eleven which are analogous to Ramanujan's forty identities for Rogers–Ramanujan functions. Furthermore, we give interesting partition-theoretic interpretation of some of the modular relations which are derived in this paper.

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Points at rational distances from the vertices of certain geometric objects

Andrew Bremner, Maciej Ulas

Journal of Number Theory > 2016 > 158 > C > 104-133

We consider various problems related to finding points in Q2 and in Q3 which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q2, and a cube or tetrahedron in Q3. In particular, as one of several results, we prove that the set of positive rational numbers a such that there exist infinitely many rational points in the plane which lie...

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Number of class polynomials of multiple eta-quotients with odd level

Mitsusada Nakata

Journal of Number Theory > 2016 > 158 > C > 316-332

We discuss class polynomials of multiple eta-quotients for an order of an imaginary quadratic field defined by using N-systems. We give an upper bound of the number of distinct class polynomials of multiple eta-quotient constructed from N-systems. Moreover we give a sufficient condition such that class polynomials are uniquely determined independent of a choice of N-system.

INFONA - science communication portal

Journal of Number Theory

Li's criterion for Epstein zeta functions, generalization of Kronecker's limit formula and the Gauss problem

On symmetries of elliptic nets and valuations of net polynomials

Pair correlation of fractional parts derived from rational valued sequences, II

The (S,{2})-Iwasawa theory

The valuative capacity of subshifts of finite type

Monogenity of totally real algebraic extension fields over a cyclotomic field

Rational products of singular moduli

On the p-rank of tame kernel of number fields

Resultants of minimal polynomials of maximal real cyclotomic extensions

Partition identities with mixed mock modular forms

Polynomial identities on eigenforms

Explicit estimates for the number of rational points of singular complete intersections over a finite field

On a problem of countable expansions

Cooper and Lam's conjecture for generalized Bell ternary quadratic forms

Spectral correspondences for Maass waveforms on quaternion groups

Generalization of Hensel's lemma: Finding the roots of p-adic Lipschitz functions

A refinement of the Dress–Scharlau theorem

Some new modular relations for the Rogers–Ramanujan type functions of order eleven with applications to partitions

Points at rational distances from the vertices of certain geometric objects

Number of class polynomials of multiple eta-quotients with odd level

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