Article
| Title: | On Popov's explicit formula and the Davenport expansion (English) |
| Author: | Yang, Quan |
| Author: | Mehta, Jay |
| Author: | Kanemitsu, Shigeru |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 869-883 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function $a_n$ with the periodic Bernoulli polynomial weight $\bar {B}_\varkappa (nx)$ and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order $0$ or $1$ gives the well-known explicit formula for respectively the partial sum or the Riesz sum of order $1$ of PNT functions. Then we may reveal the genesis of the Popov explicit formula as the integrated Davenport series with the Riesz sum of order $1$ subtracted. The Fourier expansion of the Davenport series is proved to be a consequence of the functional equation, which is referred to as the Davenport expansion. By the explicit formula for the Davenport series, we also prove that the Davenport expansion for the von Mangoldt function is equivalent to the Kummer's Fourier series up to a formula of Ramanujan and a fortiori is equivalent to the functional equation for the Riemann zeta-function. (English) |
| Keyword: | explicit formula |
| Keyword: | Davenport expansion |
| Keyword: | Kummer's Fourier series |
| Keyword: | Riemann zeta-function |
| Keyword: | functional equation |
| MSC: | 11J54 |
| MSC: | 11M41 |
| MSC: | 11N05 |
| idZBL: | Zbl 07729542 |
| idMR: | MR4632862 |
| DOI: | 10.21136/CMJ.2023.0322-22 |
| . | |
| Date available: | 2023-08-11T14:26:54Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151779 |
| . | |
| Reference: | [1] Barner, K.: On A. Weil's explicit formula.J. Reine Angew. Math. 323 (1981), 139-152. Zbl 0446.12013, MR 0611448, 10.1515/crll.1981.323.139 |
| Reference: | [2] Chakraborty, K., Kanemitsu, S., Tsukada, H.: Arithmetical Fourier series and the modular relation.Kyushu J. Math. 66 (2012), 411-427. Zbl 1334.11072, MR 3051345, 10.2206/kyushujm.66.411 |
| Reference: | [3] Chowla, S.: On some infinite series involving arithmetical functions.Proc. Indian Acad. Sci. Sect. A 5 (1937), 511-513. Zbl 0017.00505, MR 1829818 |
| Reference: | [4] Davenport, H.: On some infinite series involving arithmetical functions.Q. J. Math., Oxf. Ser. 8 (1937), 8-13. Zbl 0016.20105, 10.1093/qmath/os-8.1.8 |
| Reference: | [5] Davenport, H.: On some infinite series involving arithmetical functions. II.Q. J. Math., Oxf. Ser. 8 (1937), 313-320. Zbl 0017.39101, 10.1093/qmath/os-8.1.313 |
| Reference: | [6] Davenport, H.: Multiplicative Number Theory.Graduate Texts in Mathematics 74. Springer, New York (1980). Zbl 0453.10002, MR 0606931, 10.1007/978-1-4757-5927-3 |
| Reference: | [7] Fawaz, A. Z.: The explicit formula for $L_0(x)$.Proc. Lond. Math. Soc., III. Ser. 1 (1951), 86-103. Zbl 0042.27302, MR 0043841, 10.1112/plms/s3-1.1.86 |
| Reference: | [8] Hamburger, H.: Über einige Beziehungen, die mit der Funktionalgleichung der Riemannschen $\zeta$-Funktion äquivalent sind.Math. Ann. 85 (1922), 129-140 German \99999JFM99999 48.1214.01. MR 1512054, 10.1007/BF01449611 |
| Reference: | [9] Hardy, G. H., Littlewood, J. E.: Some problems of Diophantine approximation: The lattice-points of a right-angled triangle I., II.Proc. Lond. Math. Soc. (2) 20 (1921), 15-36 \99999JFM99999 48.0197.07. MR 1577360, 10.1112/plms/s2-20.1.15 |
| Reference: | [10] Hartman, P., Wintner, A.: On certain Fourier series involving sums of divisors.Trav. Inst. Math. Tbilissi 3 (1938), 113-118. Zbl 0018.35401 |
| Reference: | [11] Hecke, E.: Über analytische Funktionen und die Verteilung von Zahlen mod. Eins.Abh. Math. Semin. Univ. Hamb. 1 (1921), 54-76 German \99999JFM99999 48.0197.03 \99999DOI99999 10.1007/BF02940580 . MR 3069388, 10.1007/BF02940580 |
| Reference: | [12] Ingham, A. E.: The Distribution of Prime Numbers.Cambridge Tracts in Mathematics and Mathematical Physics 30. Cambridge University Press, Cambridge (1932). Zbl 0715.11045, MR 1074573 |
| Reference: | [13] Jaffard, S.: On Davenport expansions.Fractal Geometry and Applications Proceedings of Symposia in Pure Mathematics 72. AMS, Providence (2004), 273-303. Zbl 1123.28002, MR 2112109 |
| Reference: | [14] Kanemitsu, S., Ma, J., Tanigawa, Y.: Arithmetical identities and zeta-functions.Math. Nachr. 284 (2011), 287-297. Zbl 1226.11092, MR 2790889, 10.1002/mana.200710212 |
| Reference: | [15] Kanemitsu, S., Tsukada, H.: Contributions to the Theory of Zeta-Functions: The Modular Relation Supremacy.Series on Number Theory and Its Applications 10. World Scientific, Hackensack (2015),\99999DOI99999 10.1142/8711 . Zbl 1311.11003, MR 3329611 |
| Reference: | [16] Koksma, J. F.: Diophantische Approximationen.Springer, Berlin (1974), German \99999MR99999 0344200 . Zbl 0276.10015, MR 0344200 |
| Reference: | [17] Li, H., Ma, J., Zhang, W.: On some Diophantine Fourier series.Acta Math. Sin., Engl. Ser. 26 (2010), 1125-1132. Zbl 1221.11060, MR 2644050, 10.1007/s10114-010-8387-x |
| Reference: | [18] Mikolás, M.: Mellinsche Transformation und Orthogonalität bei $\zeta(s,u)$. Verallgemeinerung der Riemannschen Funktionalgleichung von $\zeta(s)$.Acta Sci. Math. 17 (1956), 143-164 German \99999MR99999 0089864 . Zbl 0073.06403, MR 0089864 |
| Reference: | [19] Patkowski, A. E.: On Popov's formula involving the von Mangoldt function.Pi Mu Epsilon J. 15 (2019), 45-47. Zbl 1441.11206, MR 4263971 |
| Reference: | [20] Patkowski, A. E.: A note on arithmetic Diophantine series.Czech. Math. J. 71 (2021), 1149-1155. Zbl 07442480, MR 4339117, 10.21136/CMJ.2021.0311-20 |
| Reference: | [21] Patkowski, A. E.: On Davenport expansions, Popov's formula, and Fine's query.Available at https://arxiv.org/abs/2004.05644v3 (2021), 8 pages. MR 4330286 |
| Reference: | [22] Patkowski, A. E.: On arithmetic series involving the fractional part function.Tsukuba J. Math. 46 (2022), 145-152. Zbl 07598531, MR 4489190, 10.21099/tkbjm/20224601145 |
| Reference: | [23] Popov, A. I.: Several series containing primes and roots of $\zeta(s)$.C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 41 (1943), 362-363. Zbl 0061.08203, MR 0010581 |
| Reference: | [24] Prachar, K.: Primzahlverteilung.Die Grundlehren der Mathematischen Wissenschaften 91. Springer, Berlin (1957), German. Zbl 0080.25901, MR 0516660 |
| Reference: | [25] Romanov, N. P.: Hilbert spaces and the theory of numbers. II.Izv. Akad. Nauk SSSR, Ser. Mat. 15 (1951), 131-152 Russian. Zbl 0044.04002, MR 0043122 |
| Reference: | [26] Segal, S. L.: On an identity between infinite series of arithmetic functions.Acta Arith. 28 (1976), 345-348. Zbl 0319.10050, MR 0387222, 10.4064/aa-28-4-345-348 |
| Reference: | [27] Srivastava, H. M., Choi, J.: Series Associated with the Zeta and Related Functions.Kluwer Academic, Dordrecht (2001). Zbl 1014.33001, MR 1849375, 10.1007/978-94-015-9672-5 |
| Reference: | [28] Titchmarsh, E. C.: Some properties of the Riemann zeta-function.Q. J. Math., Oxf. Ser. 14 (1943), 16-26. Zbl 0061.08302, MR 0008228, 10.1093/qmath/os-14.1.16 |
| Reference: | [29] Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function.Oxford University Press, Oxford (1951). Zbl 0042.07901, MR 0046485 |
| Reference: | [30] Walfisz, A. A.: On the sums of the coefficients of certain Dirichlet series.Soobshch. Akad. Nauk Gruz. SSR 26 (1961), 9-16 Russian. Zbl 0136.33204, MR 0142514 |
| Reference: | [31] Walum, H.: Multiplication formulae for periodic functions.Pac. J. Math. 149 (1991), 383-396. Zbl 0736.11012, MR 1105705, 10.2140/pjm.1991.149.383 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
