Displaying similar documents to “Convolutions and Mean Square Estimates of Certain Number-theoretic Error Terms”

On differences of two squares

Manfred Kühleitner, Werner Nowak (2006)

Open Mathematics

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The arithmetic function ρ(n) counts the number of ways to write a positive integer n as a difference of two squares. Its average size is described by the Dirichlet summatory function Σn≤x ρ(n), and in particular by the error term R(x) in the corresponding asymptotics. This article provides a sharp lower bound as well as two mean-square results for R(x), which illustrates the close connection between ρ(n) and the number-of-divisors function d(n).

An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial

Hideaki Ishikawa, Kohji Matsumoto (2011)

Open Mathematics

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We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.

On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

Manfred Kühleitner, Werner Nowak (2013)

Open Mathematics

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The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

On the riemann zeta-function and the divisor problem II

Aleksandar Ivić (2005)

Open Mathematics

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Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of ζ 1 2 + i t . If E *(t)=E(t)-2πΔ*(t/2π) with Δ * x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain ∫ 0 T E * t 5 d t ≪ ε T 2 + ε and ∫ 0 T E * t 544 75 d t ≪ ε T 601 225 + ε . It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of ζ 1 2 + i t .

Some problems on mean values of the Riemann zeta-function

Aleksandar Ivić (1996)

Journal de théorie des nombres de Bordeaux

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Several problems and results on mean values of ζ ( s ) are discussed. These include mean values of | ζ ( 1 2 + i t ) | and the fourth moment of | ζ ( σ + i t ) | for 1 / 2 &lt; σ &lt; 1 .

Nonlinear exponential twists of the Liouville function

Qingfeng Sun (2011)

Open Mathematics

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Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum ∑ X ⩽ n ⩽ 2 X λ ( n ) e 2 π i α n , 0 ≠ α ∈ ℝ The main tool we use is Vaughan’s identity for λ(n).