Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Alexander E. Patkowski

On Davenport expansions, Popov's formula and Fine's query

Pages: 59-65
Received: 19 April 2022
Accepted in revised form: 27 July 2022
Mathematics Subject Classification: 11L20, 11M06.
Keywords: Davenport expansions; Riemann zeta function; von Mangoldt function.

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Abstract: We establish an explicit connection between a Davenport expansion and the Popov sum. Asymptotic analysis follows as a result of these formulas. New solutions to a query of N. J. Fine are offered, and a proof of Davenport expansions is detailed.

P. T. Bateman and S. Chowla, Some special trigonometric series related to the distribution of prime numbers, J. London Math. Soc. 38 (1963), 372-374. MR0153639
K. Chakraborty, S. Kanemitsu and H. Tsukada, Arithmetical Fourier series and the modular relation, Kyushu J. Math. 66 (2012), no. 2, 411-427. MR3051345
H. Davenport, On some infinite series involving arithmetic function, Quart. J. Math. Oxford Ser. 8 (1937), no. 1, 8-13. DOI   Zbl
H. Davenport, On some infinite series involving arithmetical functions II, Quart. J. Math. Oxford Ser. 8 (1937), no. 1, 313-320. DOI   Zbl
H. M. Edwards, Riemann's Zeta function, Reprint of the 1974 original, Dover Publications, Mineola, NY, 2001. MR1854455
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th edition, A. Jeffrey and D. Zwillinger, eds., Elsevier/Academic Press, Amsterdam, 2007. MR2360010
S. Inoue, Relations among some conjectures on the Möbius function and the Riemann zeta-function, Acta Arith. 191 (2019), no. 1, 1-32 MR3998979
S. Jaffard, On Davenport expansions, in "Fractal geometry and applications: a jubilee of Benoît Mandelbrot", Proc. Sympos. Pure Math., 72, Part 1, Amer. Math. Soc., Providence, RI, 2004, 273-303. MR2112109
H. L. Li, J. Ma and W. P. Zhang, On some Diophantine Fourier series, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 6, 1125-1132. MR2644050
R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, 2001. MR1854469
A. E. Patkowski, On Popov's formula involving the von Mangoldt function, Pi Mu Epsilon J. 15 (2019), no. 1, 45-47. MR4263971
A. I. Popov, Several series containing primes and roots of \(\zeta(s)\), C. R. (Doklady) Acad. Sci. URSS (N.S.) 41 (1943), 362-363. MR0010581
O. Ramaré, Explicit estimates for the summatory function of \(\Lambda(n)/n\) from the one of \(\Lambda(n)\), Acta Arith. 159 (2013), no. 2, 113-122. MR3062910
S. L. Segal, On an identity between infinite series of arithmetic functions, Acta Arith. 28 (1975/76), no. 4, 345-348. MR0387222
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edition, Oxford University Press, New York, 1986. MR0882550

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