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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]

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]

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]

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 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]

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 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 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]

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]

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]

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]

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 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. [ABSTRACT FROM AUTHOR]