Elliptic surfaces to class groups and Selmer groups
Research in Number Theory, 2025, DOI Link
View abstract ⏷
In this note, we connect the n-torsions of the Picard group of an elliptic surface to the n-divisibility of the class group of torsion fields for a given integer n>1. We also connect the n-divisibility of the Selmer group to that of the class group of torsion fields.
Finiteness of Selmer groups associated to degree zero cycles on an abelian variety over a global function field
Ramanujan Journal, 2025, DOI Link
View abstract ⏷
In this note, we define the notion of Tate–Shafarevich group and Selmer group of the Chow group of zero cycles of degree zero of an abelian variety defined over a global function field and prove that the Selmer group is finite.
Shifted convolution sums of divisor functions associated with the symmetric power lifts of GL(2)-forms
Ramanujan Journal, 2025, DOI Link
View abstract ⏷
We study the average of shifted convolution sums involving the Fourier coefficients of the symmetric power lifts of Hecke eigenforms. Moreover, we obtain a similar result for the Hecke–Maass eigenforms under certain suitable conditions.
On the Product of Zeta-Functions
Mathematics, 2025, DOI Link
View abstract ⏷
In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference (Formula presented.), an integer with the Voronoĭ function weight (Formula presented.). In the case of (Formula presented.), the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The (Formula presented.) case is the divisor function, while the (Formula presented.) case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan.
On the Plus Parts of the Class Numbers of Cyclotomic Fields
Chinese Annals of Mathematics. Series B, 2025, DOI Link
View abstract ⏷
The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers. They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields. At the end, they provide some numerical examples supporting our results.
on the complete solutions of a generalized Lebesgue-Ramanujan-Nagell equation
Quaestiones Mathematicae, 2025, DOI Link
View abstract ⏷
We consider the generalized Lebesgue-Ramanujan-Nagell equation x2 + 17k 41ℓ 59m = 2δ yn in the unknown integers x ≥ 1, y > 1, n ≥ 3 and k, ℓ, m ≥ 0 satisfying gcd(x, y) = 1. We first find all the integer solutions of the above equation, and then use this result to determine all the integer solutions of some other Lebesgue-Ramanujan-Nagell type equations. Our method uses the classical results of Bilu, Hanrot and Voutier on existence of primitive divisors of Lehmer sequences in combination with number theoretic arguments and computer search.
Modular Relations and Parity in Number Theory
Infosys Science Foundation Series in Mathematical Sciences, 2025, DOI Link
View abstract ⏷
This book describes research problems by unifying and generalizing some remote-looking objects through the functional equation and the parity relation of relevant zeta functions, known as the modular relation or RHB correspondence. It provides examples of zeta functions introduced as absolutely convergent Dirichlet series, not necessarily with the Euler product. The book generalizes this to broader cases, explaining the special functions involved. The extension of the Chowla–Selberg integral formula and the Hardy transform are key, substituting the Bochner modular relation in the zeta function of Maass forms. The book also develops principles to deduce summation formulas as modular relations and addresses Chowla’s problem and determinant expressions for class numbers. Many books define zeta functions using Euler products, excluding Epstein and Hurwitz-type zeta functions. Euler products are constructed from objects with a unique factorization domain property. This book focuses on using the functional equation, called the modular relation, specifically the ramified functional equation of the Hecker type. Here, the gamma factor is the product of two gamma functions, leading to the Fourier–Whittaker expansion, and reducing to the Fourier–Bessel expansion or the Chowla–Selberg integral formula for Epstein zeta functions.
Class groups of imaginary biquadratic fields
Research in Number Theory, 2025, DOI Link
View abstract ⏷
We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng [13] and the other from Banerjee and Hoque [3].
On the hamburger-oberhettinger-soni modular relations
Mathematical Analysis: Theory and Applications, 2024, DOI Link
Preface
Springer Proceedings in Mathematics and Statistics, 2024,
ON THE EXPONENTIAL DIOPHANTINE EQUATION x2+ pmqn= 2yp
New Zealand Journal of Mathematics, 2024, DOI Link
View abstract ⏷
We study the exponential Diophantine equation x2+ pmqn= 2ypin positive integers x, y, m, n, and odd primes p and q using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation.
Generalized fruit diophantine equation and hyperelliptic curves
Monatshefte fur Mathematik, 2024, DOI Link
View abstract ⏷
We show the insolvability of the Diophantine equation axd-y2-z2+xyz-b=0 in Z for fixed a and b such that a≡1(mod12) and b=2da-3, where d is an odd integer and is a multiple of 3. Further, we investigate the more general family with b=2da-3r, where r is a positive odd integer. As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over Q. We conclude by providing some numerical evidence corroborating the main results.
Advances in algebra analysis and topology
Advances in Algebra Analysis and Topology, 2024, DOI Link
View abstract ⏷
This book presents cutting-edge research, advanced techniques, and practical applications of Algebra Analysis and Topology. It offers in-depth insights, theoretical developments, and practical applications, showcasing the richness and interdisciplinary nature of algebra, analysis, and topology. The book fosters a deeper understanding of the fundamental principles while also highlighting the latest advancements and emerging trends in these disciplines. Readers are encouraged to apply the theoretical concepts and techniques to solve mathematical problems, engaging with the book's problem-solving approach. By combining theoretical foundations, practical applications, and interdisciplinary perspectives, this book aims to inspire new avenues of research and contribute to the ongoing development of these dynamic fields. • Provides a comprehensive and accessible resource that covers a broad range of topics in algebra, analysis, and topology, understanding of the interconnections between these mathematical fields • Encompasses both classical topics and cutting-edge research areas within algebra, analysis, and topology • Covers foundational concepts, advanced theories, and their applications in diverse fields such as physics, computer science, engineering, and economics • Offers sophisticated tools and methodologies to tackle complex problems and deepen the understanding of these disciplines • Explores how algebra, analysis, and topology intersect with other fields of mathematics and how their concepts and techniques can be applied in related disciplines It serves as a valuable reference for graduate students, researchers, and mathematicians seeking to deepen their knowledge and engage with the latest advancements in these fundamental branches of mathematics.
DIOPHANTINE D(n)-QUADRUPLES IN (Formula presented) DIOFANTOVE D(n)-ČETVORKE U (Formula presented)
Glasnik Matematicki, 2024, DOI Link
View abstract ⏷
Let d be a square-free integer and Z[√d] a quadratic ring of integers. For a given n ∈ Z[√d], a set of m non-zero distinct elements in Z[√ d] is called a Diophantine D(n)-m-tuple (or simply D(n)-m-tuple) in Z[√ d] if product of any two of them plus n is a square in Z[√d]. Assume that d ≡ 2 (mod 4) is a positive integer such that x2 − dy2 = −1 and x2 − dy2 = 6 are solvable in integers. In this paper, we prove the existence of infinitely many D(n)-quadruples in Z[√d] for n = 4m + 4k√d with m, k ∈ Z satisfying m ≢ 5 (mod 6) and k ≢ 3 (mod 6). Moreover, we prove the same for n = (4m + 2) + 4k√ d when either m ≢ 9 (mod 12) and k ≢ 3 (mod 6), or m ≢ 0 (mod 12) and k ≢ 0 (mod 6). At the end, some examples supporting the existence of quadruples in Z[√ d] with the property D(n) for the above exceptional n’s are provided for d = 10.
ADVANCES IN NUMBER THEORY AND APPLIED ANALYSIS
Debnath P., Srivastava H.M., Chakraborty K., Kumam P.
Advances in Number Theory and Applied Analysis, 2023, DOI Link
View abstract ⏷
Presently, the exploration of the applications of different techniques and tools of number theory and mathematical analysis are extensively prevalent in various areas of engineering, mathematical, physical, biological and statistical sciences. This book will present the most recent developments in these two fields through contributions from eminent scientists and mathematicians worldwide. The book will present the current state of the art development in these two areas through original new contributions and surveys. As such, readers will find several useful tools and techniques to develop their skills and expertise in number theory and applied analysis. New research directions are also indicated in each of the chapters. This book is meant for graduate students, faculty and researchers willing to expand their knowledge in number theory and mathematical analysis. The readers of this book will require minimum prerequisites of analysis, topology, number theory and functional analysis.
Unification of Chowla’s Problem and Maillet–Demyanenko Determinants
Mathematics, 2023, DOI Link
View abstract ⏷
Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of (Formula presented.) (Formula presented.). On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula.
Diophantine Triples with the Property D(n) for Distinct n’s
Mediterranean Journal of Mathematics, 2023, DOI Link
View abstract ⏷
We prove that for every integer n, there exist infinitely many D(n)-triples which are also D(t)-triples for t∈ Z with n≠ t. We also prove that there are infinitely many D(- 1) -triples in Z[i] which are also D(n)-triple in Z[i] for two distinct n’s other than n= - 1 and these triples are not equivalent to any triple with the property D(1).
A Unifying Principle in the Theory of Modular Relations †
Mathematics, 2023, DOI Link
View abstract ⏷
The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function (Formula presented.) satisfying a suitable functional equation, one can generalize it to (Formula presented.) in the form of an integral involving the Mellin transform (Formula presented.) of a certain suitable function (Formula presented.) and process it further as (Formula presented.). Under the condition that (Formula presented.) is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for (Formula presented.). Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion).
Lehmer sequence approach to the divisibility of class numbers of imaginary quadratic fields
Ramanujan Journal, 2023, DOI Link
View abstract ⏷
Let k≥ 3 and n≥ 3 be odd integers, and let m≥ 0 be any integer. For a prime number ℓ, we prove that the class number of the imaginary quadratic field Q(ℓ2m-2kn) is either divisible by n or by a specific divisor of n. Applying this result, we construct an infinite family of certain tuples of imaginary quadratic fields of the form: (Q(d),Q(d+1),Q(4d+1),Q(2d+4),Q(2d+16),⋯,Q(2d+4t))with d∈ Z and 1 ≤ 4 t≤ 2 | d| whose class numbers are all divisible by n. Our proofs use some deep results about primitive divisors of Lehmer sequences.
On Joint Discrete Universality of the Riemann Zeta-Function in Short Intervals
Chakraborty K., Kanemitsu S., Laurincikas A.
Mathematical Modelling and Analysis, 2023, DOI Link
View abstract ⏷
In the paper, we prove that the set of discrete shifts of the Riemann zeta-function (ζ(s + 2πia1k), . . . , ζ(s + 2πiark)), k ∈ N, approximating analytic nonvanishing functions f1(s), . . . , fr(s) defined on {s ∈ C : 1/2 < Res < 1} has a positive density in the interval [N,N + M] with M = o(N), N → ∞, with real algebraic numbers a1, . . . , ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series.
On a Conjecture of Franušić and Jadrijević: Counter-Examples
Results in Mathematics, 2023, DOI Link
View abstract ⏷
Let d≡2(mod4) be a square-free integer such that x2- dy2= - 1 and x2- dy2= 6 are solvable in integers. We prove the existence of infinitely many quadruples in Z[d] with the property D(n) when n∈{(4m+1)+4kd,(4m+1)+(4k+2)d,(4m+3)+4kd,(4m+3)+(4k+2)d,(4m+2)+(4k+2)d} for m, k∈ Z. As a consequence, we provide few counter examples to Conjecture 1 of Franušić and Jadrijević [Math. Slovaca 69, 1263–1278 (2019)].
Certain eta-quotients and ℓ -regular overpartitions
Ramanujan Journal, 2022, DOI Link
View abstract ⏷
Let A¯ ℓ(n) be the number of overpartitions of n into parts not divisible by ℓ. In this paper, we prove that A¯ ℓ(n) is almost always divisible by pij if pi2ai≥ℓ, where j is a fixed positive integer and ℓ=p1a1p2a2⋯pmam with primes pi> 3. We obtain a Ramanujan-type congruence for A¯ 7 modulo 7. We also exhibit infinite families of congruences and multiplicative identities for A¯ 5(n).
ON SOME SYMMETRIES OF THE BASE n EXPANSION OF 1/m: THE CLASS NUMBER CONNECTION
Pacific Journal of Mathematics, 2022, DOI Link
View abstract ⏷
Suppose that m≡1 mod 4 is a prime and that n≡3 mod 4 is a primitive root modulo m. We obtain a relation between the class number of the imaginary quadratic field Q(√−nm) and the digits of the base n expansion of 1/m. Secondly, if m ≡ 3 mod 4, we study some convoluted sums involving the base n digits of 1/m and arrive at certain congruence relations involving the class number of Q(√−m) modulo certain primes p which properly divide n+1.
On the Diophantine Equation dx2+ p2aq2b= 4 yp
Results in Mathematics, 2022, DOI Link
View abstract ⏷
We investigate the solvability of the Diophantine equation in the title, where d> 1 is a square-free integer, p, q are distinct odd primes and x, y, a, b are unknown positive integers with gcd (x, y) = 1. We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.
On the Fourier coefficients of certain Hilbert modular forms
Ramanujan Journal, 2022, DOI Link
View abstract ⏷
We prove that given any ϵ> 0 , a non-zero adelic Hilbert cusp form f of weight k=(k1,k2,…,kn)∈(Z+)n and square-free level n with Fourier coefficients Cf(m) , there exists a square-free integral ideal m with N(m)≪k03n+ϵN(n)6n2+12+ϵ such that Cf(m) ≠ 0. The implied constant depends on ϵ, F.
Complex Powers of L-functions and Integers Without Small Prime Factors
Chakraborty K., Kanemitsu S., Laurincikas A.
Mediterranean Journal of Mathematics, 2022, DOI Link
View abstract ⏷
The Selberg type divisor problem (Selberg in J Indian Math Soc 18:83–87, 1954, Serre in A course in arithmetic, Springer, Berlin–Heidelberg, 1973), pertains to the study on the coefficients of the complex power of the zeta-function as has been exhibited in Banerjee et al. (Kyushu J Math 71:363–385). The original objective of Selberg was to apply the results to the problem related to the prime number theorem. However, the complex powers turn out to be of independent interest and have applications in studying the mean values of the zeta and other L-functions. We consider the zeta-function (á la Hecke) associated to the normalised cusp forms for the full modular group in the quest of continuing the research started in Laurinčikas and Steuding (Cent Eur Sci J Math 2:1–18, 2004) on the summatory function of the complex power in the form of the Dirichlet convolution of the coefficients of such zeta-functions. The convolution includes the one with the identity and we may cover the results in Laurinčikas and Steuding (Cent Eur Sci J Math 2:1–18, 2004). We treat the general case of the Selberg class zeta-functions and the modular form case is an example. It is interesting to note that one of the main roles is played by the integers without large and small prime factors which have been studied extensively. Our approach has a merit of attaining both, the result on the summatory function as well as a result entailing the classical result on distrobution of integers without small prime factors together.
On moments of non-normal number fields
Journal of Number Theory, 2022, DOI Link
View abstract ⏷
Let K be a number field over Q and let aK(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≤XaKl(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.
On a family of elliptic curves of rank at least 2
Czechoslovak Mathematical Journal, 2022, DOI Link
View abstract ⏷
Let Cm:y2 = x3 − m2x + p2q2 be a family of elliptic curves over ℚ, where m is a positive integer and p, q are distinct odd primes. We study the torsion part and the rank of Cm(ℚ). More specifically, we prove that the torsion subgroup of Cm(ℚ) is trivial and the ℚ-rank of this family is at least 2, whenever m ≢ 0 (mod 3), m ≢ 0 (mod 4) and m ≡ 2 (mod 64) with neither p nor q dividing m.
Sign changes in restricted coefficients of Hilbert modular forms
Agnihotri R., Chakraborty K., Krishnamoorthy K.
Ramanujan Journal, 2022, DOI Link
View abstract ⏷
Let f be an adelic Hilbert cusp form of weight k and level n over a totally real number field F. In this paper, we study the sign changes in the Fourier coefficients of f when restricted to square-free integral ideals and integral ideals in “arithmetic progression". In both cases we obtain qualitative results and in the former case we obtain a quantitative result as well. Our results are general in the sense that we do not impose any restriction to the totally real number field F, the weight k or the level n.
Sign changes of certain arithmetical function at prime powers
Czechoslovak Mathematical Journal, 2021, DOI Link
View abstract ⏷
We examine an arithmetical function defined by recursion relations on the sequence {f(pk)}k∈ℕ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.
Generalized hypergeometric Bernoulli numbers
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 2021, DOI Link
View abstract ⏷
We introduce generalized hypergeometric Bernoulli numbers for Dirichlet characters. We study their properties, including relations, expressions and determinants. At the end in Appendix we derive first few expressions of these numbers.
On the Diophantine Equation cx2+ p2m= 4 yn
Results in Mathematics, 2021, DOI Link
View abstract ⏷
Let c be a square-free positive integer and p a prime satisfying p∤ c. Let h(- c) denote the class number of the imaginary quadratic field Q(-c). In this paper, we consider the Diophantine equation cx2+p2m=4yn,x,y≥1,m≥0,n≥3,gcd(x,y)=1,gcd(n,2h(-c))=1,and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.
On the structure of order 4 class groups of Q(√n2+1)
Annales Mathematiques du Quebec, 2021, DOI Link
View abstract ⏷
Groups of order 4 are isomorphic to either Z/ 4 Z or Z/ 2 Z× Z/ 2 Z. We give certain sufficient conditions permitting to specify the structure of class groups of order 4 in the family of real quadratic fields Q(n2+1) as n varies over positive integers. Further, we compute the values of Dedekind zeta function attached to these quadratic fields at the point - 1. As a side result, we show that the size of the class group of this family could be made as large as possible by increasing the size of the number of distinct odd prime factors of n.
On the solutions of certain lebesgue-ramanujan-nagell equations
Rocky Mountain Journal of Mathematics, 2021, DOI Link
View abstract ⏷
We completely solve the Diophantine equation x2 +2k11l19m = yn in integers x, y ≥ 1; k, l,m ≥ 0 and n ≥ 3 with gcd(x, y) = 1, except the case 2 | k, 2]lm and 5 | n. We use this result to recover some earlier results in the same direction.
An analogue of Wilton’s formula and values of Dedekind zeta functions
Journal of Mathematical Analysis and Applications, 2021, DOI Link
View abstract ⏷
J. R. Wilton obtained an expression for the product of two Riemann zeta functions. This expression played a crucial role to find the approximate functional equation for the product of two Riemann zeta functions in the critical region. We find analogous expressions for the product of two Dedekind zeta functions and then use these expressions to find some expressions for Dedekind zeta values attached to arbitrary real as well as quadratic number fields at any positive integer.
On some recursion relations for horn’s hypergeometric functions of three variables
Agabwal P., Shehata A., Moustafaf S.I., Chakraborty K.
Proceedings of the Jangjeon Mathematical Society, 2021, DOI Link
View abstract ⏷
The principal aim of this paper to study the recursion formulas for the Horns hypergeometric functions of three variables. Earlier in [Shehata, A.; and Moustafa, S.I. Some new results for Horn's hypergeometric functions Ti and IV Journal of Mathematics and Computer Science, (2021), 23 (1), 26-35.], and Pathan et al. [Pathan, M.A.; Shehata, A.; and Moustafa, S.I. Certain new formulas for the Horns hypergeometric functions. Acta Uni-versitatis Apulensis, (2020)] have studied the new results for Horns hypergeometric functions. Motivated by the above works here we will derive some contiguous relation for the families of Horn hypergeometric functions Ga, Gb, Gc, Gdand G∗cof three variables. After that we will establish the differential reclusion relations and differential operators for Ga, Gb, Gc, Gdand G∗c, of three variables, respectively.
Class Groups of Number Fields and Related Topics
Class Groups of Number Fields and Related Topics, 2020, DOI Link
View abstract ⏷
This book gathers original research papers and survey articles presented at the “International Conference on Class Groups of Number Fields and Related Topics,” held at Harish-Chandra Research Institute, Allahabad, India, on September 4-7, 2017. It discusses the fundamental research problems that arise in the study of class groups of number fields and introduces new techniques and tools to study these problems. Topics in this book include class groups and class numbers of number fields, units, the Kummer-Vandiver conjecture, class number one problem, Diophantine equations, Thue equations, continued fractions, Euclidean number fields, heights, rational torsion points on elliptic curves, cyclotomic numbers, Jacobi sums, and Dedekind zeta values. This book is a valuable resource for undergraduate and graduate students of mathematics as well as researchers interested in class groups of number fields and their connections to other branches of mathematics. New researchersto the field will also benefit immensely from the diverse problems discussed. All the contributing authors are leading academicians, scientists, researchers, and scholars.
Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations
Publicationes Mathematicae Debrecen, 2020, DOI Link
View abstract ⏷
It is well-known that for p = 1; 2; 3; 7; 11; 19; 43; 67; 163, the class num- ber of Q(√-p) is one. We use this fact to determine all the solutions of x2+ pm= 4ynin non-negative integers x; y;m and n.
Primary rank of the class group of real cyclotomic fields
Rocky Mountain Journal of Mathematics, 2020, DOI Link
View abstract ⏷
Let H = Q.(ζn +ζn-1) and ℓ be an odd prime such that q ≅ 1 .mod ℓ/ for a prime factor q of n. We get a bound on the ℓ-rank of the class group of H in terms of the ℓ-rank of the class group of a real quadratic subfield contained in H. At the end we look into few numerical examples.
Preface
Class Groups of Number Fields and Related Topics, 2020,
Distribution of generalized mex-related integer partitions
Hardy-Ramanujan Journal, 2020, DOI Link
View abstract ⏷
The minimal excludant or “mex” function for an integer partition π of a positive integer n, mex(π), is the smallest positive integer that is not a part of π. Andrews and Newman introduced σmex(n) to be the sum of mex(π) taken over all partitions π of n. Ballantine and Merca generalized this combinatorial interpretation to σrmex(n), as the sum of least r-gaps in all partitions of n. In this article, we study the arithmetic density of σ2 mex(n) and σ3 mex(n) modulo 2k for any positive integer k.
A note on certain real quadratic fields with class number up to three
Kyushu Journal of Mathematics, 2020, DOI Link
View abstract ⏷
We obtain criteria for the class number of certain Richaud–Degert type real quadratic fields to be three. We also treat a couple of families of real quadratic fields of Richaud–Degert type that were not considered earlier, and obtain similar criteria for the class number of such fields to be two and three.
Exponent of Class Group of Certain Imaginary Quadratic Fields
Czechoslovak Mathematical Journal, 2020, DOI Link
View abstract ⏷
Let n > 1 be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form ℚ(x2−2yn) whose ideal class group has an element of order n. This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups.
Divisibility of Selmer groups and class groups
Hardy-Ramanujan Journal, 2019, DOI Link
View abstract ⏷
In this paper, we study two topics. One is the divisibility problem of class groups of quadratic number fields and its connections to algebraic geometry. The other is the construction of Selmer group and Tate-Shafarevich group for an abelian variety defined over a number field.
Asymptotic behaviour of a Lambert series à la Zagier: Maass case
Ramanujan Journal, 2019, DOI Link
View abstract ⏷
Hafner and Stopple proved a conjecture of Zagier on the asymptotic expansion of a Lambert series involving Ramanujan’s tau function with the main term involving the nontrivial zeros of the Riemann zeta function. Recently, Chakraborty et. al. have extended this result to any cusp form over the full modular group and also over any congruence subgroup. The aim here is to study the asymptotic behaviour of a similar Lambert series involving the coefficients of Maass cusp forms over the full modular group.
Divisibility of class numbers of certain families of quadratic fields
Journal of the Ramanujan Mathematical Society, 2019,
View abstract ⏷
We construct some families of quadratic fields whose class numbers are divisible by 3. The main tools used are a trinomial introduced by Kishi and a parametrization of Kishi and Miyake of a family of quadratic fields whose class numbers are divisible by 3. At the end we compute class number of these fields for some small values and verify our results.
Pell-type equations and class number of the maximal real subfield of a cyclotomic field
Ramanujan Journal, 2018, DOI Link
View abstract ⏷
We investigate the solvability of the Diophantine equation x2- my2= ± p in integers for certain integer m and prime p. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number is strictly larger than 1.
Divisibility of the class numbers of imaginary quadratic fields
Chakraborty K., Hoque A., Kishi Y., Pandey P.P.
Journal of Number Theory, 2018, DOI Link
View abstract ⏷
For a given odd integer n>1, we provide some families of imaginary quadratic number fields of the form Q(x2−tn) whose ideal class group has a subgroup isomorphic to Z/nZ.
An asymptotic expansion of a Lambert series associated to cusp forms
Chakraborty K., Juyal A., Kumar S.D., Maji B.
International Journal of Number Theory, 2018, DOI Link
View abstract ⏷
Zagier's conjecture on the asymptotic expansion of the Lambert series Σn=1∞∞2(n)exp(-nz), where ∞(n) is the Ramanujan's tau function, was proved by Hafner and Stopple. Recently, Chakraborty, Kanemitsu and Maji have extended this result to any cusp forms over the full modular group. The goal of this paper is to extend the asymptotic behavior to cusp forms over any congruence subgroup of the full modular group.
Divisibility of class numbers of quadratic fields: qualitative aspects
Trends in Mathematics, 2018, DOI Link
View abstract ⏷
Class numbers of quadratic fields have been the object of attention for many years, and there exist a large number of interesting results. This is a survey aimed at reviewing results concerning the divisibility of class numbers of both real and imaginary quadratic fields. More precisely, to review the question ‘do there exist infinitely many real (respectively imaginary) quadratic fields whose class numbers are divisible by a given integer?’ This survey also contains the current status of a quantitative version of this question.
Pairs of integers which are mutually squares
Chakraborty K., Jimenez Urroz J., Pappalardi F.
Science China Mathematics, 2017, DOI Link
View abstract ⏷
We derived an asymptotic formula for the number of pairs of integers which are mutually squares. Earlier results dealt with pairs of integers subject to the restriction that they are both odd, co-prime and squrefree. Here we remove all these restrictions and prove (similar to the best known one with restrictions) that the number of such pair of integers upto a large real X is asymptotic to cX2logX with an absolute constant c which we give explicitly. Our error term is also compatible to the best known one.
Abel–Tauber process and asymptotic formulas
Banerjee D., Chakraborty K., Kanemitsu S., Maji B.
Kyushu Journal of Mathematics, 2017, DOI Link
View abstract ⏷
The Abel–Tauber process consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration–differentiation process. In this article, we use the Abel–Tauber process to establish an interesting asymptotic expansion for the Riesz sums of arithmetic functions with best possible error estimate. The novelty of our paper is that we incorporate the Selberg type divisor problem in this process by viewing the contour integral as part of the residual function. The novelty also lies in the uniformity of the error term in the additional parameter which varies according to the cases. Generalization of the famous Selberg Divisor problem to arithmetic progression has been made by Rieger [Zum Teilerproblem von Atle Selberg. Math. Nachr. 30 (1965), 181–192], Marcier [Sums of the form Σg(n)/f (n). Canad. Math. Bull. 24 (1981), 299–307], Nakaya [On the generalized division problem in arithmetic progressions. Sci. Rep. Kanazawa Univ. 37 (1992), 23–47] and around the same time Nowak [Sums of reciprocals of general divisor functions and the Selberg division problem, Abh. Math. Sem. Univ. Hamburg 61 (1991), 163–173] studied the related subject of reciprocals of an arithmetic function and obtained an asymptotic formula with the Vinogradov–Korobov error estimate with the main term as a finite sum of logarithmic terms. We shall also elucidate the situation surrounding these researches and illustrate our results by rich examples.
Quadratic reciprocity and some “non-differentiable” functions
Trends in Mathematics, 2017, DOI Link
View abstract ⏷
Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. In [28] (Proc Int Conf–Number Theory 1, 107–116, 2004), Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta transformation formula and Gerver (Amer J Math 92, 33–55, 1970) [12] was the first to give a proof of differentiability/non-differentiability of Riemann’s function. The aim here is to survey some of the work done in these two directions and concentrates more onto a recent work of the first author along with Kanemitsu and Li (Res Number Theory 1, 14, 2015) [5]. In that work (Kanemitsu and Li, Res Number Theory 1, 14, 2015) [5], an integrated form of the theta function was utilised and the advantage of that is that while the theta function Θ (τ) is a dweller in the upper half-plane, its integrated form F(z) is a dweller in the extended upper half-plane including the real line, thus making it possible to consider the behaviour under the increment of the real variable, where the integration is along the horizontal line.
Modular-type relations associated to the Rankin–Selberg L-function
Ramanujan Journal, 2017, DOI Link
View abstract ⏷
Hafner and Stopple proved a conjecture of Zagier relating to the asymptotic behaviour of the inverse Mellin transform of the symmetric square L-function associated with the Ramanujan tau function. In this paper, we prove a similar result for any cusp form over the full modular group.
Ewald expansions of a class of zeta-functions
SpringerPlus, 2016, DOI Link
View abstract ⏷
The incomplete gamma function expansion for the perturbed Epstein zeta function is known as Ewald expansion. In this paper we state a special case of the main formula in Kanemitsu and Tsukada (Contributions to the theory of zeta-functions: the modular relation supremacy. World Scientific, Singapore, 2014) whose specifications will give Ewald expansions in the H-function hierarchy. An Ewald expansion for us are given by (Formula presented.) or its variants. We shall treat the case of zeta functions which satisfy functional equation with a single gamma factor which includes both the Riemann as well as the Hecke type of functional equations and unify them in Theorem 2. This result reveals the H-function hierarchy: the confluent hypergeometric function series entailing the Ewald expansions. Further we show that some special cases of this theorem entails various well known results, e.g., Bochner–Chandrasekharan theorem, Atkinson–Berndt theorem etc.
Additive functions on the greedy and lazy fibonacci expansions
Journal of Integer Sequences, 2016,
View abstract ⏷
We find all complex-valued functions that are additive with respect to both the greedy and the lazy Fibonacci expansions. We take it a little further by considering the subsets of these functions that are also multiplicative. In the final section we extend these ideas to Tribonacci expansions.
Quadratic reciprocity and Riemann’s non-differentiable function
Research in Number Theory, 2015, DOI Link
View abstract ⏷
Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. Here we provide a combined proof of both the facts. In (Proc. Int. Conf.–NT 1;107–116, 2004) Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta-transformation formula and Gerver (Amer. J. Math. 92;33–55, 1970) was the first to give a proof of differentiability/non-differentiabilty of Riemnan’s function. We use an integrated form of the theta function and the advantage of that is that while the theta-function Θ(τ) is a dweller in the upper-half plane, its integrated form F(z) is a dweller in the extended upper half-plane including the real line, thus making it possible to consider the behavior under the increment of the real variable, where the integration is along the horizontal line. 2010 Mathematics Subject Classification: Primary: 11A15, Secondary: 11F27
On power moments of the Hecke multiplicative functions
Journal of the Australian Mathematical Society, 2015, DOI Link
View abstract ⏷
In a recent paper, Soundararajan has proved the quantum unique ergodicity conjecture by getting a suitable estimate for the second order moment of the so-called 'Hecke multiplicative' functions. In the process of proving this he has developed many beautiful ideas. In this paper we generalize his arguments to a general kth power and provide an analogous estimate for the kth power moment of the Hecke multiplicative functions. This may be of general interest.
Preventing Unknown Key-Share Attack using Cryptographic Bilinear Maps
Journal of Discrete Mathematical Sciences and Cryptography, 2014, DOI Link
View abstract ⏷
Here we add a third pass to the two pass AK protocols, MTI/AO protocol and a two-pass protocol proposed by L. Law et al. using cryptographic bilinear maps. The added third pass provides additional key confirmation and also prevents the unknown key-share attack which could be successfully launched on the above mentioned two-pass protocols. © 2014 © Taru Publications.
On partial sums of a spectral analogue of the Möbius function
Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 2013, DOI Link
View abstract ⏷
Sankaranarayanan and Sengupta introduced the function μ*(n) corresponding to the Möbius function. This is defined by the coefficients of the Dirichlet series 1/L f (s), where L f (s) denotes the L-function attached to an even Maaß cusp form f . We will examine partial sums of μ*(n). The main result is ∑n≤x μ*(n) = O(x exp(-Avlog x)), where A is a positive constant. It seems to be the corresponding prime number theorem. © Indian Academy of Sciences.
Arithmetical fourier series and the modular relation
Kyushu Journal of Mathematics, 2012, DOI Link
View abstract ⏷
We consider the zeta functions satisfying the functional equation with multiple gamma factors and prove a far-reaching theorem, an intermediate modular relation, which gives rise to many (including many of the hitherto found) arithmetical Fourier series as a consequence of the functional equation. Typical examples are the Diophantine Fourier series considered by Hardy and Littlewood and one considered by Hartman and Wintner, which are reciprocals of each other, in addition to our previous work. These have been thoroughly studied by Li, Ma and Zhang. Our main contribution is to the effect that the modular relation gives rise to the Fourier series for the periodic Bernoulli polynomials and Kummer's Fourier series for the log sin function, thus giving a foundation for a possible theory of arithmetical Fourier series based on the functional equation. © 2012 Faculty of Mathematics, Kyushu University.
A stamped blind signature scheme based on elliptic curve discrete logarithm problem
International Journal of Network Security, 2012,
View abstract ⏷
Here we present a stamped blind digital signature scheme which is based on elliptic curve discrete logarithm prob-lem and collision-resistant cryptographic hash functions.
The modular relation and the digamma function
Kyushu Journal of Mathematics, 2011, DOI Link
View abstract ⏷
In this paper we shall locate a class of fundamental identities for the gamma function and trigonometric functions in the chart of functional equations for the zetafunctions as a manifestation of the underlying modular relation. We use the beta-transform but not the inverse Heaviside integral. Instead we appeal to the reciprocal relation for the Euler digamma function which gives rise to the partial fraction expansion for the cotangent function. Through this we may incorporate basic results from the theory of the digamma (and gamma) function, thereby dispensing also with the beta-transform. Section 4 could serve as a foundation of the theory of the gamma function through the digamma function. © 2011 Faculty of Mathematics, Kyushu University.
On the chowla-selberg integral formula for non-holomorphic eisenstein series
Integral Transforms and Special Functions, 2010, DOI Link
View abstract ⏷
In this note, we get the Fourier expansion for the non-holomorphic Eisenstein series by slight modification of Maass' original method, which enables us to prove as a bonus, two integral representations for the modified Bessel functionof the third kind.This kind revealsahidden inner structureofthe non-holomorphic Eisenstein series and the Bessel diffenrential equation. We also explain a work of Motohashi on the Kronecker limit formula for the Epstein zeta-function from our point of view. © 2010 Taylor & Francis.
On the values of a class of dirichlet series at rational arguments
Proceedings of the American Mathematical Society, 2010, DOI Link
View abstract ⏷
In this paper we shall prove that the combination of the general distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations between the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor's gamma functions, among which is the Gauss second formula for the digamma function. © 2009 American Mathematical Society.
Manifestations of the Parseval identity
Chakraborty K., Kanemitsu S., Li J., Wang X.
Proceedings of the Japan Academy Series A: Mathematical Sciences, 2009, DOI Link
View abstract ⏷
In this paper, we make structural elucidation of some interesting arithmetical identities in the context of the Parseval identity. In the continuous case, following Romanoff [R] and Wintner [Wi], we study the Hilbert space of square-integrable functions L2(0,1) and provide a new complete orthonormal basis-the Clausen system-, which gives rise to a large number of intriguing arithmetical identities as manifestations of the Parseval identity. Especially, we shall refer to the identity of Mikolás'-Mordell. Secondly, we give a new look at enormous number of elementary mean square identities in number theory, including H. Walum's identity [Wa] and Mikolás' identity (1.16). We show that some of them may be viewed as the Parseval identity. Especially, the mean square formula for the Dirichlet L-function at 1 is nothing but the Parseval identity with respect to an orthonormal basis constructed by Y. Yamamoto [Y] for the linear space of all complex-valued periodic functions. © 2009 The Japan Academy.
Vistas of special functions II
Vistas of Special Functions II, 2009, DOI Link
View abstract ⏷
This book (Vista II), is a sequel to Vistas of Special Functions (World Scientific, 2007), in which the authors made a unification of several formulas scattered around the relevant literature under the guiding principle of viewing them as manifestations of the functional equations of associated zeta-functions. In Vista II, which maintains the spirit of the theory of special functions through zeta-functions, the authors base their theory on a theorem which gives some arithmetical Fourier series as intermediate modular relations — avatars of the functional equations. Vista II gives an organic and elucidating presentation of the situations where special functions can be effectively used. Vista II will provide the reader ample opportunity to find suitable formulas and the means to apply them to practical problems for actual research. It can even be used during tutorials for paper writing.
Class numbers with many prime factors
Journal of Number Theory, 2008, DOI Link
View abstract ⏷
Here, we construct infinitely many number fields of any given degree d > 1 whose class numbers have many prime factors. © 2008 Elsevier Inc. All rights reserved.
Exponents of class groups of real quadratic fields
International Journal of Number Theory, 2008, DOI Link
View abstract ⏷
In this paper, we show that the number of real quadratic fields K of discriminant ΔK < χ whose class group has an element of order g (with g even) is ge; x1/g/5 if x > x0, uniformly for positive integers g ≤ (log log x)/ (8 log log log x). We also apply the result to find real quadratic number fields whose class numbers have many prime factors. © 2008 World Scientific Publishing Company.
Exponents of class groups of real quadratic function fields (II)
Proceedings of the American Mathematical Society, 2006, DOI Link
View abstract ⏷
Let g be an even positive integer. We show that there are ≫ q l/g/l2 polynomials D ∈ scirpt F signq[t] with deg(D) ≤ l such that the ideal class group of the real quadratic extensions script F signq(t, √) have an element of order g. © 2005 American Mathematical Society.
Exponents of class groups of real quadratic function fields
Proceedings of the American Mathematical Society, 2004, DOI Link
View abstract ⏷
We show that there are ≫ ql/(2g) polynomials D ∈ double-struck F signq[t] with deg(D) ≤ l such that the ideal class group of the real quadratic extensions double-struck F signq(t, √D) has an element of order g.
On the number of real quadratic fields with class number divisible by 3
Proceedings of the American Mathematical Society, 2003, DOI Link
View abstract ⏷
We find a lower bound for the number of real quadratic fields whose class groups have an element of order 3. More precisely, we establish that the number of real quadratic fields whose absolute discriminant is ≤ x and whose class group has an element of order 3 is ≫ x5/6 improving the existing best known bound ≫ x1/6 of R. Murty.
On the number of fourier coefficients that determine a hilbert modular form
Proceedings of the American Mathematical Society, 2002, DOI Link
View abstract ⏷
We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.
Modular forms which behave like theta series
Mathematics of Computation, 1997, DOI Link
View abstract ⏷
In this paper, we determine all modular forms of weights 36 ≤ k ≤ 56, 4 | k, for the full modular group SL2(ℤ) which behave like theta series, i.e., which have in their Fourier expansions, the constant term 1 and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in ℝ3 (resp. in ℝ4) for the cases k = 36, 40 and 44 (resp. for the cases k = 48, 52 and 56). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.
A note on Jacobi forms of higher degree
Chakraborty K., Ramakrishnan B., Vasudevan T.C.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1995, DOI Link
A note on Hecke eigenforms
Archiv der Mathematik, 1994, DOI Link
On the average behaviour of an arithmetical function
Archiv der Mathematik, 1994, DOI Link
