Asian Journal of Probability and Statistics

On Propositions Pertaining to the Riemann Hypothesis II

Pathikrit Basu *

20128 White Cloud Circle, West Linn, Oregon, USA.

*Author to whom correspondence should be addressed.

---

Abstract

Aims/Objectives: In this paper, we define certain classes of non-zeroes of the Riemann zeta function. We also present associated algorithms for finding these non-zeroes, which can enable corresponding computations. Some theoretical connections are also drawn with mixed integer programming and continuous Diophantine approximation. We also study, for points in the domain of the Riemann zeta function, their induced distributions over the unit circle.

Keywords: Probability measures over the unit circle, Riemann hypothesis, Riemann zeta function, complex variables, complex functions

---

---

---

References

Basu P. On the Riemann Hypothesis. Asian Journal of Probability and Statistics. 2022;20(4):51{56.

Basu P. On Propositions Pertaining to the Riemann Hypothesis. Asian Journal of Probability and Statistics. 2023;21(2):16-21.

Riemann, B. "On the number of primes less than a given magnitude". Monatsberichte der Berliner Akademie 1859:1{10.

Riemann B. Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse. Ges. Math. Werke und Wissenschaftlicher Nachla. 1859; 2:145{155.

Stein, E, Shakarchi, R. Complex analysis. Princeton University Press; 2010.

Schrijver, A. Theory of linear and integer programming. John Wiley Sons; 1998.

Conforti, M, Cornuejols, G, Zambelli, G, Conforti, M, Cornuejols, G, Zambelli, G. Integer programming models. Springer; 2014.

Minkowski, H. Diophantine approximations: An introduction to number theory. Mathematical lectures at the Universitat Gottingen; 1907.

Sprindzhuk, V. "Metric theory of Diophantine approximations". (No Title) 1979.

Pollington, A, Vaughan, R. "The k-dimensional Duffin and Schaeffer conjecture". Mathematika 1990; 37(2):190{200.

Schmidt, W. Diophantine approximation. Springer Science Business Media; 1996.

Queffelec, H, Queffelec, M, Queffelec. Diophantine approximation and Dirichlet series. Springer; 2013.

Koukoulopoulos, D, Maynard, J. "On the duffin-schae er conjecture". Annals of mathematics 2020; 192(1):251{307.

Mangoldt H. Zur Verteilung der Nullstellen der Riemannschen Funktion......". Mathematische Annalen. 1905;61:1{19.

Hardy, G. Sur les zeros de la fonction ζ (s) de Riemann. CR Acad. Sci. Paris 1914;158:1012.

Hardy, G, Littlewood, J. The zeros of Riemann's zeta-function on the critical line. Mathematische Zeitschrift. 1921;10(3):283{317.

Conrey, J. "The riemann hypothesis". Notices of the AMS 2003; 50(3):341{353.

Lagarias J. An elementary problem equivalent to the Riemann hypothesis". The American Mathematical Monthly. 2002;109(6):534{543.

Bump, D, Choi, KK, Kurlberg, P, Vaaler, J. "A local Riemann hypothesis, I". Mathematische Zeitschrift 2000; 233(1):1{18.

Borwein, Peter, Choi, Stephen, Rooney, Brendan, Weirathmueller, Andrea, A. The Riemann hypothesis: a resource for the acionado and virtuoso alike. Springer; 2008.

Platt, D, Trudgian, T. "The Riemann hypothesis is true up to 3_ 10 12". Bulletin of the London Mathematical Society 2021; 53(3):792{797.

Nicolas, JL. "The sum of divisors function and the Riemann hypothesis". The Ramanujan Journal 2021:1{45.

Johnston, D. "Improving bounds on prime counting functions by partial veri cation of the Riemann hypothesis". The Ramanujan Journal 2022; 59(4):1307{1321.

Gram, JP. "Note sur les zeros de la fonction ζ (s) de Riemann". Acta Mathematica 1903.

Turing, A. "Some calculations of the Riemann zeta-function". Proceedings of the London Mathematical Society 1953; 3(1):99{117.

Ramanujan, S. "New expressions for Riemann's functions (s) and (t)". Quart. J. Math 1915; 46:253{260.

Pierpont, J. Functions of a complex variable. Ginn; 1914.

Narici L. Beckenstein, E. Topological vector spaces. Chapman and Hall/CRC; 2010.

Weyl, H. Uber die gleichverteilung von zahlen mod. eins". Mathematische Annalen. 1916; 77(3):313{352.

Kuipers, L, Niederreiter, H. Uniform distribution of sequences. Courier Corporation; 2012.

Glyn Harman. Diophantine approximation by prime numbers. Journal of the London Mathematical Society.1991;2(2):218{226.