Contributions to the Theory of Zeta-Functions

Abstract. In this paper we shall locate a class of fundamental identities for the gamma function and trigonometric functions in the chart of functional equations for the zetafunctions as a manifestation of the underlying modular relation. We use the beta-transform but not the inverse Heaviside integral. Instead we appeal to the reciprocal relation for the Euler digamma function which gives rise to the partial fraction expansion for the cotangent function. Through this we may incorporate basic results from the theory of the digamma (and gamma) function, thereby dispensing also with the beta-transform. Section 4 could serve as a foundation of the theory of the gamma function through the digamma function.

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“…Factoring out (z)/z and substituting (14), what remains is e · e −z · e z−1 = 1 and the result follows.…”

Section: Proposition 1 the Weierstrass Product Representation For Thmentioning

confidence: 91%

“…However, our recent theory of general modular relations (cf. [8,9,14]), which gives identities equivalent to the functional equation…”

Section: Introduction and The Digamma Functionmentioning

confidence: 99%

The Modular Relation and the Digamma Function

2011

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“…Hardy and Riesz [13,22,30,Chapter 6] and references therein. The generating function F (s) in (2.2) is to be chosen as the essential factor Z * (s).…”

Section: Riesz Sumsmentioning

confidence: 99%

Abel-Tauber Process and Asymptotic Formulas

et al. 2017

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Abstract. The Abel-Tauber process consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration-differentiation process. In this article, we use the Abel-Tauber process to establish an interesting asymptotic expansion for the Riesz sums of arithmetic functions with best possible error estimate. The novelty of our paper is that we incorporate the Selberg type divisor problem in this process by viewing the contour integral as part of the residual function. The novelty also lies in the uniformity of the error term in the additional parameter which varies according to the cases. Generalization of the famous Selberg Divisor problem to arithmetic progression has been made by Rieger [163][164][165][166][167][168][169][170][171][172][173] studied the related subject of reciprocals of an arithmetic function and obtained an asymptotic formula with the Vinogradov-Korobov error estimate with the main term as a finite sum of logarithmic terms. We shall also elucidate the situation surrounding these researches and illustrate our results by rich examples.

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“…Restoration and extrapolation of band-limited signals, Kayanomori 20, 10-14. Takahashi, K., Matsuzaki, T. , Hirano, G. , Kaida, T., Fujio, M., and Kanemitsu, S. (2014 …”

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confidence: 97%

“…Takahashi, K., Matsuzaki, T. , Hirano, G. , Kaida, T., Fujio, M., and Kanemitsu, S. (2014). Restoration and extrapolation of band-limited signals, Kayanomori 20, 10-14.…”

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confidence: 99%

Back Matter

2016

A Quick Introduction to Complex Analysis

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Contributions to the Theory of Zeta-Functions

Cited by 20 publications

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The Modular Relation and the Digamma Function

The Modular Relation and the Digamma Function

Abel-Tauber Process and Asymptotic Formulas

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