Search

We study the behavior of the shifted convolution sum involving even power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the k -full kernel function for any fixed integer k ≥ 2. [ABSTRACT FROM AUTHOR]

Suppose that x is a sufficiently large number and j ≥ 2 is any integer. Let L (s , sym j f) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL 2 (Z) . Also, let λ sym j f (n) be the n-th normalized Dirichlet coefficient of L (s , sym j f) . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients λ sym j f (n) over two sparse sequences of positive integers, which improves previous results. [ABSTRACT FROM AUTHOR]

As a consequence of their work, Bruce C. Berndt, Ronald J. Evans, Larry Joel Goldstein and Michael Razar obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. We give a short proof of it and also express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums. [ABSTRACT FROM AUTHOR]

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree. [ABSTRACT FROM AUTHOR]

Let χ denote a primitive, non-quadratic Dirichlet character with conductor q , and let L (s , χ) denote its associated Dirichlet L -function. We show that | L (1 , χ) | ≥ 1 / (9.12255 log ⁡ (q / π)) for sufficiently large q , and that | L (1 , χ) | ≥ 1 / (9.69030 log ⁡ (q / π)) for all q ≥ 2 , improving some results of Louboutin. The improvements come from an averaging argument that simplifies Louboutin's approach, and the construction of certain trigonometric polynomials using simulated annealing, a technique often used in applied mathematics and computer science. We highlight the benefits of this approach to a pure mathematical problem. [ABSTRACT FROM AUTHOR]

Let q ≥ 2 be an integer and let s q (n) be the sum-of-digitsfunction of n in base q. The function s q (n) has been studied in many directions and many properties have been obtained on the distribution of s q (n) and s q (P (n)) , where P is a suitable polynomial. In this paper we derive the generatingfunctions of s p (n d mod p k) for d ≥ 2 and prime p ≥ 2 by using the properties of Dirichlet character, and study the correlation properties of the sequences ((- 1) n 2 mod p k ) n < p k . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism. [ABSTRACT FROM AUTHOR]

In [4] , B. C. Berndt and A. Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, in a follow-up paper [20] , S. Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified K -Bessel function in light of recent results obtained by the first author and B. Maji [1]. Moreover, we provide a new expression for L (1 , χ) from which we establish the positivity of L (1 , χ) for any real primitive character χ. In addition, we deduce Cohen-type identities and then exhibit the Voronoï-type summation formulas for them. [ABSTRACT FROM AUTHOR]

In this article, we prove an asymptotic formula for the fourth power mean of a general s -dimensional hyper-Kloosterman sum. We find the number of solutions of certain congruence equations mod p which play an integral part to prove our main result. We use estimates for character sums and analytic methods to prove our theorem. [ABSTRACT FROM AUTHOR]

Menon's identity is a classical identity involving gcd sums and the Euler totient function ϕ . In a recent paper, Zhao and Cao (Int. J. Number Theory13(9) (2017) 2373–2379) derived the Menon-type identity ∑ k = 1 n (k - 1 , n) χ (k) = ϕ (n) τ (n d) , where χ is a Dirichlet character mod n with conductor d. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao–Cao identity can be improved if one uses the method we employ here. [ABSTRACT FROM AUTHOR]

We prove a lower bound for the number of conductors q ≤ Q with q ≡ 1 (mod r) having t prime divisors of Dirichlet characters χ of fixed order r and with given values χ (p) for primes p ≤ B. We also give an elliptic curve variant of this result. [ABSTRACT FROM AUTHOR]

In this article, we prove an asymptotic formula for the fourth power mean of a general 4-dimensional Kloosterman sum. We use a result of P. Deligne, which counts the number of F p -points on the surface (x - 1) (y - 1) (z - 1) (1 - x y z) - u x y z = 0 , u ≠ 0 , and then take an average of the error terms over u to prove the asymptotic formula. We also find the number of solutions of certain congruence equations mod p which are used to prove our main result. [ABSTRACT FROM AUTHOR]

Let r be a positive integer ≥2. We consider a family of primitive Dirichlet characters of order r with conductor co-prime to r. For this family, we compute the one-level density with explicit lower order terms in two ways, using Weil's explicit formula and the Ratios conjecture. Also, the n -level density for the family twisted by a fixed cuspidal automorphic representation π of G L M (A Q) is obtained. It turns out that, when r ≥ 3 , the symmetry type for our family is always unitary. [ABSTRACT FROM AUTHOR]

Let χ be a Dirichlet character modulo a prime p. We give explicit upper bounds on q 1 < q 2 < ... < q n , the n smallest prime nonresidues of χ. More precisely, given n 0 and p 0 there exists an absolute constant C = C (n 0 , p 0) > 0 such that q n ≤ C p 1 4 (log ⁡ p) n + 1 2 whenever n ≤ n 0 and p ≥ p 0. [ABSTRACT FROM AUTHOR]

Let q ≥ 3 be an integer, χ denote a Dirichlet character modulo q , for any real number a ≥ 0 , we define the generalized Dirichlet L -functions L (s , χ , a) = ∑ n = 1 ∞ χ (n) (n + a) s , where s = σ + i t with σ > 1 and t both real. It can be extended to all s by analytic continuation. In this paper, we study the mean value properties of the generalized Dirichlet L -functions, and obtain several sharp asymptotic formulae by using analytic method. [ABSTRACT FROM AUTHOR]

Abstract We give a new proof of the fact that the growth (with respect to n) of the p -adic valuation of h n , 2 − is linear, where h n , 2 − denotes the minus part of the (S , { 2 }) -refined class number of the cyclotomic field Q (μ p n + 1 ) , as defined by Hu and Kim in [1]. As a consequence of our proof, we obtain an explicit relation between the p -adic valuation of h n , 2 − and the p -adic valuation of h n − , the minus part of the class number h n of the cyclotomic field Q (μ p n + 1 ). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we seek to evaluate the most general form of moments of L (1 , χ) twisted by character value χ (c) as χ ranges over non-principal Dirichlet characters mod f. And we use this result to model the joint distribution for χ (c) and L (1 , χ). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic methods, the solutions of the congruence equation mod p and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 3-dimensional Kloostermann sums mod p, and give a sharp asymptotic formula for it. [ABSTRACT FROM AUTHOR]

The classical Menon's identity [P. K. Menon, On the sum ∑ (a − 1 , n) [ (a , n) = 1 ] , J. Indian Math. Soc. (N.S.) 29 (1965) 155–163] states that ∑ a ∈ ℤ n ∗ gcd (a − 1 , n) = φ (n) σ 0 (n) , where for a positive integer n , ℤ n ∗ is the group of units of the ring ℤ n = ℤ / n ℤ , gcd (,) represents the greatest common divisor, φ (n) is the Euler's totient function and σ k (n) = ∑ d | n d k is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: ∑ a ∈ ℤ n ∗ b 1 , ... , b k ∈ ℤ n gcd (a − 1 , b 1 , ... , b k , n) χ (a) = φ (n) σ k n d , where k is a non-negative integer and χ is a Dirichlet character modulo n whose conductor is d. Our result can be viewed as an extension of Zhao and Cao's result [Another generalization of Menon's identity, Int. J. Number Theory13(9) (2017) 2373–2379] to k > 0. It can also be viewed as an extension of Sury's result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo58 (2009) 99–108] to Dirichlet characters. [ABSTRACT FROM AUTHOR]

In this work, we make an extension of earlier results provided by Kubilus and Linnik who modelled specific class of Brownian motions using the sums [1], where p is the sequence of odd square free numbers. Our results are relevant and can be used in a range of applications, especially when applying Monte-Carlo methods in finance and econometrics. [ABSTRACT FROM AUTHOR]

Let be a Dirichlet character (mod ) with conductor . In a quite recent paper Zhao and Cao deduced the identity , which reduces to Menon's identity if is the principal character (mod ). We generalize the above identity by considering even functions (mod ), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums. [ABSTRACT FROM AUTHOR]

In this paper, by applying the p-adic q-integrals to a family of continuous differentiable functions on the ring of p-adic integers, we construct new generating functions for generalized Apostol-type numbers and polynomials attached to the Dirichlet character of a finite abelian group. By using these generating functions with their functional equations, we derive various new identities and relations for these numbers and polynomials. These results are generalizations of known identities and relations including some well-known families of special numbers and polynomials such as the generalized Apostol-type Bernoulli, the Apostol-type Euler, the Frobenius-Euler numbers and polynomials, the Stirling numbers, and other families of numbers and polynomials. Moreover, by the help of these generating functions, we also construct other new families of numbers and polynomials with their generating functions. By using these functions, we investigate some fundamental properties of these numbers and polynomials. Finally, we also give explicit formulas for computing the Apostol-Bernoulli and Apostol-Euler numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy the same topological property consisting of possessing zeros arbitrarily near every vertical line contained in some substrips inside their critical strip. We also show that the real projections of the zeros of the partial sums of the alternating zeta function, for some particular cases, are dense in their critical intervals. [ABSTRACT FROM AUTHOR]

Let be the Euler's totient function and be the number of divisors of . Menon's beautiful identity states that Here we extend this identity to Dirichlet characters mod . [ABSTRACT FROM AUTHOR]

In this article we estimate the number of integers up to X which can be properly represented by a positive-definite, binary, integral quadratic form of small discriminant. This estimate follows from understanding the vector of signs that arises from computing the Legendre symbol of small integers n at multiple primes. [ABSTRACT FROM AUTHOR]

We use analytic methods to obtain an explicit formula for the fourth power mean where , is a Dirichlet character modulo and denotes the summation over all such that . This extends the result of Chen, Ai and Cai by overcoming the limitation . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 2-dimensional Kloostermann sums mod p, and give an exact computational formula for it. [ABSTRACT FROM AUTHOR]

Given c , a positive integer, we give an asymptotic formula for ∑ χ ≠ χ 0 χ ( c ) | L ( 1 , χ ) | 2 , where χ is the non-trivial Dirichlet character mod f with f coprime to c . [ABSTRACT FROM AUTHOR]

Let {Fn}n≥0 be the sequence of Fibonacci numbers. The aim of this paper is to give linear independence results over Q(√5) for the infinite series Σn=1∞ χj(n)/Fn with certain nonprincipal real Dirichlet characters χj. We also deduce the irrationality results for the special principal Dirichlet characters and for other multiplicative functions. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct some families of generalized Apostol-type special numbers and polynomials attached to the Dirichlet character. Using the p-adic q-Volkenborn integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we give generating functions for these numbers and polynomials. These numbers and polynomials are associated with some well-known special numbers and polynomials such as the Peters polynomials, the Boole polynomials, the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, the generalized Apostol-Daehee numbers and polynomials, the Stirling numbers and the Bernoulli numbers of the first kind. We investigate some properties of these numbers and polynomials with their generating functions. Using these generating functions and their functional equation, we derive some identities and relations including some special numbers and polynomials. Finally, we give p-adic q-Volkenborn integral representations for these numbers and polynomials with combinatorial sums. [ABSTRACT FROM AUTHOR]

We consider a collection ofL-functions of elliptic curves twisted by a Dirichlet character moduloq(qis a prime number), and prove for this collection a joint limit theorem for weakly convergent probability measures in the space of analytic functions asq→ ∞. The limit measure is given explicitly. [ABSTRACT FROM PUBLISHER]

Structural theorems for spaces of modular forms with respect to congruence subgroups are proved. The Dedekind η-function plays an important role in our study. [ABSTRACT FROM AUTHOR]

A limit theorem involving an increasing modulus of the character is obtained for twists with the Dirichlet character of L-functions of elliptic curves. [ABSTRACT FROM AUTHOR]

Recall that a Dirichlet character is called imprimitive if it is induced from a character of smaller level, and otherwise it is called primitive. In this paper, we introduce a modification of "inducing to higher level" which causes imprimitive characters to behave primitively, in the sense that the properties of the associated Gauss sum and the functional equation of the attached L-function take on a form usually associated to a primitive character. [ABSTRACT FROM AUTHOR]

In this paper, we use evaluations of Gauss sums modulo pk to derive expressions that allow us for a given generalized Markoff-Hurwitz equation to determine the number of its solutions over ℤ/pkℤ if the number of solutions over ℤ/pℤ is known. We also calculate the corresponding Poincaré series. [ABSTRACT FROM AUTHOR]

We introduce Dirichlet's type of twisted Eulerian polynomials by using p-adic fermionic q-invairant integral in the p-adic integer ring and obtain some new interesting identities. Using a complex contour integral representation based on the generating function of Dirichlet's type of twisted Eulerian polynomials, we get L-type function which interpolates for Dirichlet's type of Eulerian polynomials at negative integers. [ABSTRACT FROM AUTHOR]

For the sum of pairwise products of values Zm (s;δ1/γ-,0)Zm (s;0,δ2/γ) of the Hecke zeta-functions over the ring of Gaussian integers, we obtain a formula for the Laplace transform on the half-line. [ABSTRACT FROM AUTHOR]

In the paper, a limit theorem for weakly convergent probability measures on the complex plane for twisted with Dirichlet characterL-functions of elliptic curves with an increasing modulus of the character is proved. [ABSTRACT FROM PUBLISHER]

Using elementary methods, we obtain an explicit formula for the fourth power meanΣm=1q'Σχmodq|Σa=1q'χ(a)e(mak+naq)|4 for arbitrary positive integer k, where e(y)=e2πiy, χ is a Dirichlet character modulo q and Σa=1q' denotes the summation over all a such that (a,q)=1. This improves H.N. Liu's result by averting the restriction (k,q)=1. [ABSTRACT FROM AUTHOR]

In this paper, we prove a multidimensional limit theorem for moduli of twists ofL-functions of elliptic curves. The limit measure in this theorem is defined by the characteristic transforms. [ABSTRACT FROM PUBLISHER]

We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums. [ABSTRACT FROM AUTHOR]

Abstract: We use elementary methods to obtain an explicit formula for the fourth power mean where , χ is a Dirichlet character modulo q and denotes the summation over all a with . This extends a result of Z.F. Xu et al. by avoiding the restriction . [Copyright &y& Elsevier]

Abstract: Let χ be a non-principal Dirichlet character modulo a prime p. Let denote the two smallest prime non-residues of χ. We give explicit upper bounds on that improve upon all known results. We also provide a good upper estimate on the product which has an upcoming application to the study of norm-Euclidean Galois fields. [Copyright &y& Elsevier]

Let q be a sufficiently large integer and χ be a Dirichlet character modulo q. In this paper, we extend the product Π L(1, χ) with prime q, arising from the Kummer conjecture, to the products of some general Dirichlet series, and give some meaningful estimates for them. [ABSTRACT FROM AUTHOR]

In the paper, a limit theorem for the argument of twisted with Dirichlet character L-functions of elliptic curves with an increasing modulus of the character is proved. [ABSTRACT FROM PUBLISHER]

We establish a numerically explicit version of the Póólya--Vinogradov inequality for the sum of values of a Dirichlet character on an interval. While the technique of proof is essentially that of Landau from 1918, the result we obtain has better constants than in other numerically explicit versions that have been found more recently. [ABSTRACT FROM AUTHOR]

For integers m, n, q, k, with q, k≧1 and Dirichlet characters $\chi, \chi' \text {\rm \;(mod}\,q)$ we define a generalized Kloosterman sum with a Dirichlet character and a Gauss sum G( a, χ′) as coefficient, where e( z)= e. The aim of this paper is to study the fourth power mean obtaining explicit formulas for M( q). [ABSTRACT FROM AUTHOR]