On some specific order estimates for the Mertens function and its relation to the non-trivial zeros of ζ(s)
Subham Desubham581994@gmail.comDepartment of MathematicsHauz KhasIndian Institute of Technology DelhiNew Delhi, Delhi, 110016, India0009-0001-3265-4354View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Feb 2025
- Published Online:
- 12 Mar 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2414
- Pages:
- 2757–2772
Abstract
This article presents a brief overview of the Mertens function M(n) and Redheffer matrices An, deriving specific order estimates for M(n) to investigate the distribution of non-trivial zeros of the Riemann zeta function ζ(s). Using these tools, we establish a necessary and sufficient condition for the validity of the Riemann Hypothesis.
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References
[1] William W. Barrett, Rodney W. Forcade, and Andrew D. Follington, “On the spectral radius of a (0,1) matrix related to the Mertens function,” Linear Algebra and Its Applications, vol. 107, pp. 151–159 (1987).
[2] Edward Charles Titchmarsh, The Theory of the Riemann Zeta-Function. Oxford, U.K.: Oxford University Press, 1951; 2nd ed., revised by David Rodney Heath-Brown, Oxford, U.K.: Oxford University Press (1986).
[3] Bernhard Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse,” Monatsberichte der Königlichen Preußischen Akademie der Wissenschaften zu Berlin, pp. 671–680 (Nov. 1859). [Online]. Available: https://www.claymath.org/sites/default/files/ezeta.pdf
[4] Lars Valerian Ahlfors, Complex Analysis, 3rd ed. New York, NY, USA: McGraw-Hill Education (1979).
[5] Franz Mertens, “Über eine zahlentheoretische Funktion,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, vol. 106, no. IIa, pp. 761–830 (1897).
[6] Robert Daublebsky von Sterneck, “Die zahlentheoretische Funktion σ(n) bis zur Grenze 5 000 000,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, vol. 121, no. IIa, pp. 1083–1096 (1912).
[7] Helge von Koch, “Sur la distribution des nombres premiers,” Acta Mathematica, vol. 24, no. 1, pp. 159–182 (1901), doi: 10.1007/BF02403071.
[8] Lowell Schoenfeld, “Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II,” Mathematics of Computation, vol. 30, no. 134, pp. 337–360 (1976), doi: 10.2307/2005976.
[9] Tyler Joseph Jarvis, An Investigation into the Eigenvalues of a Matrix of Redheffer, Undergraduate Thesis, Brigham Young University, Provo, UT, USA (1989).
[10] Mario Allegri, José Luis López-Bonilla, and P. S. Kota Reddy, “Results on Riemann zeta functions using compositions,” Journal of Interdisciplinary Mathematics, vol. 27, pp. 1437–1447 (2024).
[11] Tyler Joseph Jarvis, “A dominant negative eigenvalue of a matrix of Redheffer,” Linear Algebra and Its Applications, vol. 142, pp. 141–152 (Dec. 1990), doi: 10.1016/0024-3795(90)90262-B.
[12] John S. Maybee, D. Dale Olesky, Pauline van den Driessche, and Gary Wiener, “Matrices, digraphs, and determinants,” SIAM Journal on Matrix Analysis and Applications, vol. 10, no. 4, pp. 500–519 (Oct. 1989).
[13] Wayne W. Barrett and Tyler Joseph Jarvis, “Spectral properties of a matrix of Redheffer,” Linear Algebra and Its Applications, vols. 162–164, pp. 673–683 (Feb. 1992), doi: 10.1016/0024-3795(92)90401-U.
[14] Albert Edward Ingham, The Distribution of Prime Numbers. Cambridge, U.K.: Cambridge University Press, 1932; reprinted by Stechert-Hafner, New York, NY, USA, 1964; reprinted with a foreword by R. C. Vaughan, Cambridge, U.K.: Cambridge University Press (1990).
[15] Gerhard Neubauer, “Eine empirische Untersuchung zur Mertensschen Funktion,” Numerische Mathematik, vol. 5, pp. 1–13 (1963).
[16] Henri Cohen and François Dress, “Calcul numérique de M(x),” in Rapport de l’ATP A12311 ‘Informatique 1975’, Paris, France: CNRS, pp. 11–13 (1979).
[17] François Dress, “Fonction sommatoire de la fonction de Möbius. I. Majorations expérimentales,” Experimental Mathematics, vol. 2, pp. 89–98 (1993).
[18] Andrew Michael Odlyzko and Herman J. J. te Riele, “Disproof of the Mertens conjecture,” Journal für die Reine und Angewandte Mathematik, vol. 357, pp. 138–160 (1985).
[19] János Pintz, “An effective disproof of the Mertens conjecture,” Astérisque, vols. 147–148, pp. 325–333 (1987).
[20] Wolfgang Bernhard Jurkat, “Eine Bemerkung zur Vermutung von Mertens,” Nachrichten der Österreichischen Mathematischen Gesellschaft, Sondernummer: Österreichischer Mathematikerkongress, p. 11 (1961).
[21] Wolfgang Bernhard Jurkat, “On the Mertens conjecture and related general Ω-theorems,” in Analytic Number Theory, Harold G. Diamond, Ed. Providence, RI, USA: American Mathematical Society, pp. 147–158 (1973).
[22] Raymond Spira, “Zeros of sections of the zeta function. II,” Mathematics of Computation, vol. 22, pp. 163–173 (1966).
[23] Wolfgang Bernhard Jurkat and Alfred Peyerimhoff, “A constructive approach to Kronecker approximation and its applications to the Mertens conjecture,” Journal für die Reine und Angewandte Mathematik, vols. 286–287, pp. 332–340 (1976).
[24] Herman J. J. te Riele, “Computations concerning the conjecture of Mertens,” Journal für die Reine und Angewandte Mathematik, vols. 311–312, pp. 356–360 (1979).
[25] Hans Möller, Zur Numerik der Mertens’schen Vermutung, Ph.D. dissertation, University of Ulm, Ulm, Germany (1987).
[26] Richard J. Anderson, “On the Möbius sum function,” Acta Arithmetica, vol. 59, pp. 205–213 (1991).
[27] Irving John Good and Richard Frank Churchhouse, “The Riemann hypothesis and pseudorandom features of the Möbius function,” Mathematics of Computation, vol. 22, pp. 857–861 (1968).
[28] Mohamed El Marraki, Majorations effectives de la fonction sommatoire de la fonction de Möbius, Ph.D. dissertation, University of Bordeaux, Bordeaux, France (1991).
[29] Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie. Berlin, Germany: VEB Deutscher Verlag der Wissenschaften (1963).
[30] Kevin Ford, “Vinogradov’s integral and bounds for the Riemann zeta function,” Proceedings of the London Mathematical Society, vol. 85, no. 3, pp. 565–633 (2002).
[31] Richard Hackel, “Zur elementaren Summierung gewisser zahlentheoretischer Funktionen,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, vol. 118, no. IIa, pp. 1019–1034 (1909).
[32] Robert A. MacLeod, “A new estimate for the sum M(x)=∑n≤x μ(n),” Acta Arithmetica, vol. 13, pp. 49–59, 1967; erratum, ibid., vol. 16, pp. 99–100 (1969).
[33] François Dress, “Majorations de la fonction sommatoire de la fonction de Möbius,” Bulletin de la Société Mathématique de France, Suppl., Mém. 49–50, pp. 47–52 (1977).
[34] Harold G. Diamond and Kevin S. McCurley, “Constructive elementary estimates for M(x),” in Analytic Number Theory, M. I. Knopp, Ed., Lecture Notes in Mathematics, vol. 899. Berlin, Germany: Springer, pp. 239–253 (1982).
[35] Nuno Costa Pereira, “Elementary estimate for the Chebyshev function ψ(x) and the Möbius function M(x),” Acta Arithmetica, vol. 52, pp. 307–337 (1989).
[36] François Dress and Mohamed El Marraki, “Fonction sommatoire de la fonction de Möbius. II. Majorations asymptotiques élémentaires,” Experimental Mathematics, vol. 2, pp. 99–112 (1993).
[37] Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2. Leipzig, Germany: B. G. Teubner, 1909; reprinted, New York, NY, USA: Chelsea Publishing Company (1953).
[38] Edmund Landau, Vorlesungen über Zahlentheorie, vol. 2 of 3. Leipzig, Germany: S. Hirzel Verlag (1927).
[39] Herman J. J. te Riele, “Some historical and other notes about the Mertens conjecture and its recent disproof,” Nieuw Archief voor Wiskunde, ser. 3, vol. 4, pp. 237–243 (1985).
[40] Herman J. J. te Riele, “On the history of the function M(x)/x since Stieltjes,” in Thomas Jan Stieltjes: Collected Papers, vol. 1, G. van Dijk, Ed. Berlin, Germany: Springer, pp. 69–79 (1993).
[41] Andrew Michael Odlyzko and Herman J. J. te Riele, “Disproof of the Mertens conjecture,” Journal für die Reine und Angewandte Mathematik, vol. 357, pp. 138–160 (1985).
[42] Tadej Kotnik and Jan van de Lune, “On the order of the Mertens function,” Experimental Mathematics, vol. 13, pp. 473–481 (2004).
[43] Raymond M. Redheffer, “Eine explizit lösbare Optimierungsaufgabe,” Internationale Schriftenreihe zur Numerischen Mathematik, vol. 36 (1977).
[44] Bachir Saffari, “Sur la fausseté de la conjecture de Mertens,” with an observation by Paul Lévy, Comptes Rendus de l’Académie des Sciences, ser. A, vol. 271, pp. 1097–1101 (1970).
[45] Robert Daublebsky von Sterneck, “Empirische Untersuchung über den Verlauf der zahlentheoretischen Funktion σ(n)im Intervalle von 0 bis 150000,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, vol. 106, no. IIa, pp. 835–1024, 1897.
[46] Robert Daublebsky von Sterneck, “Empirische Untersuchung über den Verlauf der zahlentheoretischen Funktion σ(n) im Intervalle von 150000 bis 500000,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, vol. 106, no. IIa, pp. 1053–1102 (1901).
[47] Masayoshi Yorinaga, “Numerical investigation of sums of the Möbius function,” Mathematical Journal of Okayama University, vol. 21, pp. 41–47 (1979).
[48] Wilhelmus M. Lioen and Jan van de Lune, “Systematic computations on Mertens’ conjecture and Dirichlet’s divisor problem by vectorized sieving,” in From Universal Morphisms to Megabytes: A Baayen Space Odyssey, K. Apt, L. Schrijver, and N. Temme, Eds. Amsterdam, The Netherlands: CWI, pp. 421–432 (1994).
[49] Frank Roesler, “Riemann’s hypothesis as an eigenvalue problem,” Linear Algebra and Its Applications, vol. 81, pp. 153–198 (1986).
[50] Tom M. Apostol, Introduction to Analytic Number Theory. New York, NY, USA: Springer-Verlag (1976).
