Estimations of psi function and harmonic numbers

Elezović, Neven

doi:10.1016/j.amc.2015.02.008

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Introduction and Summary Of Results1

Theorem 3 the Following Identities Hold: A)1

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Cited by 5 publications

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“…Further variations and extensions are given, for example, in [7] and [4], and in other references listed in these papers. One natural extension is the replacement of H nγ by ψ(x +1), where ψ(x) is the digamma function Γ (x)/Γ (x), since H nγ = ψ(n + 1).…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

Jameson

2019

J Inequal Appl

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Let H n = n r=1 1/r and H n (x) = n r=1 1/(r + x). Let ψ(x) denote the digamma function. It is shown that H n (x) + ψ(x + 1) is approximated by 1 2 log f (n + x), where f (x) = x 2 + x + 1 3 , with error term of order (n + x)-5. The cases x = 0 and n = 0 equate to estimates for H n-γ and ψ(x + 1) itself. The result is applied to determine exact bounds for a remainder term occurring in the Dirichlet divisor problem.

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Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

Jameson

2019

J Inequal Appl

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“…f) There are constants ‫ܥ‬ ଵ (for instance ݁ ିଶఊ ) and ‫ܥ‬ ଶ (for instance 3 ିଵ ) (see [18] or [19]) such that…”

Section: Theorem 3 the Following Identities Hold: A)mentioning

confidence: 99%

On values of the psi function

Adam

Piątek

Pleszczyński

et al. 2017

J. Appl. Math. Comput. Mech.

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Abstract. In the present paper, values of the psi function for many arguments connected with the golden ratio and Fibonacci numbers are determined or given in alternative form. Moreover, some integral representation of the psi function is found. This is a potential calculation base of values of the psi function for powers of argument. We also note that this integral representation gives better numerical estimation of values of the psi function than the respective Legendre's integral formula.MSC 2010: 33E50

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Approximation formulas and inequalities for the Euler-Mascheroni constant

Chen

2021

RACSAM

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Estimations of psi function and harmonic numbers

Cited by 5 publications

References 10 publications

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

On values of the psi function

Approximation formulas and inequalities for the Euler-Mascheroni constant

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