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In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris  360 (2022) 419–424]. [ABSTRACT FROM AUTHOR]

Yıldırım obtained an asymptotic formula of the discrete moment of | ζ (1 2 + i t) | over the zero of the higher derivatives of Hardy's Z -function. We give a generalization of his result on Hardy's Z -function. [ABSTRACT FROM AUTHOR]

Let q ≥ 3 be an integer, and let χ denote a Dirichlet character modulo q. For any real number a ≥ 0 , we define the generalized Dirichlet L -function as 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 using analytic continuation. For any integer m , the famous Gauss sum G (m , χ) is defined as G (m , χ) = ∑ a = 1 q χ (a) e a m q , where e (y) = e 2 π i y . This paper uses analytic methods to study the mean value properties of the generalized Dirichlet L -functions with the weight of the Gauss sums, and a sharp asymptotic formula is obtained. [ABSTRACT FROM AUTHOR]

We follow a paper by Sedunova regarding Vaughan's basic mean value Theorem to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by Cojocaru and Murty. As an application we derive a basic mean value theorem for the von Mangoldt generalized functions. [ABSTRACT FROM AUTHOR]

This paper provides a mean value theorem for arithmetic functions f defined by f (n) = ∏ d | n g (d) , where g is an arithmetic function taking values in (0 , 1 ] and satisfying some generic conditions. As an application of our main result, we prove that the density μ (n) (respectively, ρ q (n)) of normal (respectively, primitive) elements in the finite field extension 𝔽 q n of 𝔽 q are arithmetic functions of (nonzero) mean values. [ABSTRACT FROM AUTHOR]

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of k positive integers, where k ≥ 2 is fixed. We refine and generalize an asymptotic formula of Bordellès [Mean values of generalized gcd-sum and lcm-sum functions, J. Integer Seq. 10 (2007) Article ID:07.9.2, 13 pp], and extend certain related results of Hilberdink and Tóth [On the average value of the least common multiple of k positive integers, J. Number Theory 169 (2016) 327–341]. We also formulate some conjectures and open problems. [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]

In this paper, we consider the mean-value estimate for nonlinear Weyl sums over primes in short intervals and establish the related Bombieri-type theorems. These results have applications in additive problems with prime variables in short intervals. [ABSTRACT FROM AUTHOR]

In this paper, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and a generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012. [ABSTRACT FROM AUTHOR]

Let be the number of divisors of , let denote the error term in the classical Dirichlet divisor problem, and let denote the Riemann zeta-function. Several upper bounds for integrals of the type are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for and a divisor problem II, Indag. Math. 26(5) (2015) 842-866], where asymptotic formulas for were established for the above integral. [ABSTRACT FROM AUTHOR]

In this paper, we shall consider differences between the "discrete" and the "continuous" mean value formulas for any number-theoretic error term. We first derive new type formulas of these differences, which are independent from the periodic Bernoulli function. As a direct consequence of this result, we shall give the alternative proofs of previous results in the case of the Dirichlet divisor and the circle problems, and improve the previous results of the differences in higher power cases. Furthermore, we apply our present result to the case of Rankin-Selberg's series. [ABSTRACT FROM AUTHOR]

In this paper, we establish an asymptotic formula for ∑D∈ℋ2g+2 L(1, χD) as g → ∞ (fixed odd q), where ℋ2g+2 is the subset of 픽q[T] consisting of monic square-free polynomials of degree 2g + 2. As an application, we obtain a formula for the average of the class number times the regulator of the associated rings when D is taken over ℋ2g+2 as g → ∞. [ABSTRACT FROM AUTHOR]

We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length T1/11-∊. Our result generalizes a formula of Hughes and Young concerning the fourth moment of the Riemann zeta function. [ABSTRACT FROM AUTHOR]

We investigate the solubility of the congruence xy ≡ 1 ( p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums. [ABSTRACT FROM AUTHOR]

For each integer n ≥ 2, let be the index of composition of n, where γ(n) ≔ ∏p∣np. The index of composition of an integer measures the multiplicity of its prime factors. In this paper, we obtain a new asymptotic formula of the sum ∑n≤xλ-k(n). Furthermore, we improve the error term under the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

An asymptotic formula for the sum ∑ L(1, χ) is established for a family of hyperelliptic curves of genus g over a fixed finite field 픽q as g → ∞ making use of the analog of the approximate functional equation for such L-functions. As a corollary, we obtain a formula for the average of the class number of the associated rings . [ABSTRACT FROM AUTHOR]

We give a new bound for the large sieve inequality with power moduli qk that is uniform in k. The proof uses a new theorem due to Wooley from his work [Vinogradov's mean value theorem via efficient congruencing, to appear in Ann. of Math.] on efficient congruencing. [ABSTRACT FROM AUTHOR]

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that there exist infinitely many primes of the form p = [nc] such that p + 2 = Pr, where r is the least positive integer satisfying certain inequalities. In particular for $1 < c \leq \frac{30}{29} = 1.0344 \ldots$ we have r = 5. This result constitutes an improvement upon that of T. P. Peneva. [ABSTRACT FROM AUTHOR]

This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujan's paper in 1918, following Carmichael's mean-value-based concept from 1932 up to 1994. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on Ramanujan expansions. In particular, very short proofs of theorems for additive and multiplicative functions going far beyond previous borders are presented, and Ramanujan expansions that formerly have been considered mysterious are explained. [ABSTRACT FROM AUTHOR]

It is shown that Watt's new mean value theorem on sums of character sums can be included in the method described in the author's recent work [6] to show that the number of Carmichael numbers up to x exceeds x⅓ for all large x. This is done by comparing the application of Watt's original version of his mean value theorem [8] to the problem of primes in short intervals [3] with the problem of finding "small" primes in an arithmetic progression. [ABSTRACT FROM AUTHOR]

We obtain new upper bounds for the mean squared modulus of sums ∑n∈ℕAnχ(n), where the sequence (An) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (cm) satisfying cm = 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has An = ∑uvm = n α(u)β(v)cm (n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as logq(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers. [ABSTRACT FROM AUTHOR]