Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line
214 W Jennifer Lane, #6
Palatine, IL 60067
United States
E-mail: art.kawalec@gmail.com
Received:
Received: 06 November 2019; revised: 26 December 2019; accepted: 27 December 2019; published online: 31 December 2019
DOI: 10.12921/cmst.2019.0000047
Abstract:
In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an alternating and non-alternating series representation of the Riemann zeta function. Consequently, if the asymptotic limit of the harmonic series is known, then we obtain the Euler-Mascheroni constant with log(k). We further numerically compute these series for different non-trivial zeros. We also investigate a recursive formula for non-trivial zeros.
Key words:
Euler-Mascheroni constant, harmonic series, non-trivial zeros, Riemann zeta function
References:
[1] H.M. Edwards, Riemann’s Zeta Function, Dover Publication, Mineola, New York, 1974.
[2] J. Havil, Gamma: Exploring Euler’s Constant, Princeton University Press, 2003.
[3] A. Ivi´c, The Riemann Zeta-Function: Theory and Applications, Dover Publication, Mineola, New York, 1985.
[4] LMFDB- The L-functions and Modular Forms Database, http://www.lmfdb.org/, 2019.
[5] M. Wolf, 6+ infinity new expressions for the Euler-Mascheroni constant, math.NT/1904.09855, 2019.
In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an alternating and non-alternating series representation of the Riemann zeta function. Consequently, if the asymptotic limit of the harmonic series is known, then we obtain the Euler-Mascheroni constant with log(k). We further numerically compute these series for different non-trivial zeros. We also investigate a recursive formula for non-trivial zeros.
Key words:
Euler-Mascheroni constant, harmonic series, non-trivial zeros, Riemann zeta function
References:
[1] H.M. Edwards, Riemann’s Zeta Function, Dover Publication, Mineola, New York, 1974.
[2] J. Havil, Gamma: Exploring Euler’s Constant, Princeton University Press, 2003.
[3] A. Ivi´c, The Riemann Zeta-Function: Theory and Applications, Dover Publication, Mineola, New York, 1985.
[4] LMFDB- The L-functions and Modular Forms Database, http://www.lmfdb.org/, 2019.
[5] M. Wolf, 6+ infinity new expressions for the Euler-Mascheroni constant, math.NT/1904.09855, 2019.
