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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]

Inspired by two entries published in Ramanujan's lost notebook on p. 355, Berndt, Kim, and Zaharescu presented Riesz sum identities for the twisted divisor sums. Subsequently, Kim derived analogous results by replacing twisted divisor sums with twisted sums of divisor functions. In a recent work, the authors of the present paper deduced Cohen-type identities and Voronoï summation formulas associated with these twisted sums of divisor functions. The present paper aims to derive an equivalent version of the results offered in the previous article in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, the authors provide an identity for r 6 (n) , which is analogous to Hardy's famous result where r 6 (n) denotes the number of representations of natural number n as a sum of six squares. [ABSTRACT FROM AUTHOR]

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]

In this paper, we defined a new generalized Cochrane sum with Dirichlet characters, and gave the upper bound of the generalized Cochrane sum with Dirichlet characters. Moreover, we studied the asymptotic estimation problem of the mean value of the generalized Cochrane sum with Dirichlet characters and obtained a sharp asymptotic formula for it. By using this asymptotic formula, we also gave the mean value of the generalized Dedekind sum.

Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well.

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L (1 , χ) and L (1 + i t , χ) ) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions. [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]

Let d0 be a given square-free integer. We give an explicit formula Md0 (p) = A(d0)(1+ Bd0 (p)/p) for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the non-primitive odd Dirichlet characters modulo d0p induced by the odd Dirichlet characters modulo an odd prime p. Here d0 7→ A(d0) is an explicit multiplicative arithmetic function and Bd0 (f) is a twisted sum over the divisors d of d0 of Dedekind sums s(h, df). To prove that f 7→ Bd0 (f) is d0-periodic, we find a new and closed formula for the Dedekind sums f 7→ s(a+bf, c+df), for fixed integers a, b, c and d. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields. [ABSTRACT FROM AUTHOR]

In [4], the first and second authors studied the signatures of Dirichlet characters and elliptic curves. In this paper we improve their lower bound estimates in the case of elliptic curves. We also introduce and investigate integral and interval d-constant signatures for cusp forms, and prove lower bound results for the number of twisted cusp forms having positive or negative signatures. [ABSTRACT FROM AUTHOR]

For every positive integer n, Sita Ramaiah's identity states that ∑ a 1 , a 2 , a 1 + a 2 ∈ (Z / n Z) ∗ gcd (a 1 + a 2 - 1 , n) = ϕ 2 (n) σ 0 (n) where ϕ 2 (n) = ∑ a 1 , a 2 , a 1 + a 2 ∈ (Z / n Z) ∗ 1 , where (Z / n Z) ∗ is the multiplicative group of units of the ring Z / n Z and σ s (n) = ∑ d ∣ n d s . This identity can also be viewed as a generalization of Menon's identity. In this article, we generalize this identity to an algebraic number field K involving a Dirichlet character χ . Our result is a further generalization of recent results in Ji and Wang (Ramanujan J 53:585–594, 2020) and Sury (Rend Circ Mat Palermo 58:99–108, 2009). [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]

In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.

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]

Recently, the authors gave asymptotic formulas for L(s, χ), which associated with a primitive Dirichlet character χ, in terms of the generalized Bernoulli numbers. In this paper, based on the aforementioned asymptotic formulas we describe a method for calculating Dirichlet L-functions that can be used to validate the generalized Riemann hypothesis up to a height T > 0. Our method is a refinement of the one presented by Davies and Haselgrove. [ABSTRACT FROM AUTHOR]

This paper studies Menon–Sury's identity in a general case, i.e., the Menon–Sury's identity involving Dirichlet characters in residually finite Dedekind domains. By using the filtration of the ring D / n and its unit group U (D / n) , we explicitly compute the following two summations: ∑ a ∈ U (D / n) b 1 , ... , b r ∈ D / n N (⟨ a - 1 , b 1 , b 2 , ... , b r ⟩ + n) χ (a) and ∑ a 1 , ... , a s ∈ U (D / n) b 1 , ... , b r ∈ D / n N (⟨ a 1 - 1 , ... , a s - 1 , b 1 , b 2 , ... , b r ⟩ + n) χ 1 (a 1) ⋯ χ s (a s) , where D is a residually finite Dedekind domain and n is a nonzero ideal of D , N (n) is the cardinality of quotient ring D / n , χ i (1 ≤ i ≤ s) are Dirichlet characters mod n with conductor d i . [ABSTRACT FROM AUTHOR]

We give all solutions (D, G, F) of the equation G(n) = F(n² - 1) + D for every n ∊ N, where G, F are completely multiplicative functions and D ∊ C. [ABSTRACT FROM AUTHOR]

Let χ be a nonprincipal Dirichlet character modulo a prime number p ≽ 3 and let a X : = 1 2 (1 − X (− 1)) . Define the mean value ℳ p (− s , X) : = 2 p − 1 ∑ ψ (mod p) ψ (− 1) = − 1 L (1 , ψ) L (− s , X ψ ¯) (σ : = ℜ s > 0). We give an identity for ℳ p (− s , X) which, in particular, shows that ℳ p (− s , X) = L (1 − s , X) + a X 2 p s L (1 , X) ζ (− s) + o (1) (p → ∞) for fixed 0 < σ < 1 2 and | t : = J s | = o (p (1 − 2 σ) / (3 + 2 σ) ) . [ABSTRACT FROM AUTHOR]

We obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, we obtain the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus Q ∈ F q [ T ] , and the asymptotic limit is as deg Q ⟶ ∞ . This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It is also the function field q-analogue of the work of Conrey, who obtained the fourth moment of derivatives of the Riemann zeta-function. [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]

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]

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]

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]

Recently, Zhang and Lv find an asymptotic formula for the fourth power mean of the general 3-dimensional Kloosterman sum mod p. In this article we prove an improvement of their asymptotic formula. [ABSTRACT FROM AUTHOR]

Let n be a positive integer. Arai-Carlitz's identity is ∑ a , b , a + b ∈ (Z / n Z) × gcd (a + b - 1 , n) = X (n) τ (n) , where X (n) = ∑ a , b , a + b ∈ (Z / n Z) × 1 and τ (n) is the divisor function. In this paper, we generalize Arai-Carlitz's identity to the ring of algebraic integers concerning Dirichlet characters. In particular, let n be a positive integer and χ a Dirichlet character modulo n with the conductor d. Then ∑ a , b , a + b ∈ (Z / n Z) × gcd (a + b - 1 , n) χ (a) = μ (d) φ (n 0 2 / d) X (n / n 0) τ (n / d) , where n 0 | n such that n 0 has the same prime factors with d and gcd (n 0 , n n 0) = 1 , φ (n) is the Euler's totient function and μ (n) is the Möbius function. [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 X be a sufficiently great real number and M denote the set of natural numbers not exceeding X which cannot be written as a sum of a prime and a fixed degree of a prime number from the arithmetical progression with difference d. Let Ed(X) = cardM. We obtain a new numerical degree estimate for the set Ed(X) and an estimate from below for the number of presentations of n ∉ M in the specified type. The proven estimates refine the generalization for an arithmetical progression of results earlier got by V.A. Plaksin. [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]

In this paper we establish why the p-adic zeta function has a Dirichlet series expansion. We compute an improved expansion, which allows us to express it as a power-series modulo pn. Using this expansion, we compute all the zeros of Lp(s, χωj) for those quadratic characters χ of conductor < 200. For the calculation we use a PARI-GP Program. [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, 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]

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]

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 consider a collection of L-functions of elliptic curves twisted by a Dirichlet character modulo q (q is a prime number), and prove for this collection a joint limit theorem for weakly convergent probability measures in the space of analytic functions as q → ∞. The limit measure is given explicitly.

Let m 1 , ⋯ , m r be nonnegative integers, and set: M r = m 1 + ⋯ + m r . In this paper, first we establish an explicit linear decomposition of: ∏ i = 1 r B m i ( x ) m i ! in terms of Bernoulli polynomials B k ( x ) with 0 ≤ k ≤ M r . Second, for any integer q ≥ 2 , we study the mean values of the Dirichlet L-functions at negative integers: ∑ χ 1 , ⋯ , χ r ( mod q ) ; χ 1 ⋯ χ r = 1 ∏ i = 1 r L ( − m i , χ i ) where the summation is over Dirichlet characters χ i modulo q. Incidentally, a part of our work recovers Nielsen’s theorem, Nörlund’s formula, and its generalization by Hu, Kim, and Kim.

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]