PHDTHESIS Sadegh Nazardonyavi A Class of Equivalent Problems Related to the Riemann Hypothesis Tese submetida `aFaculdade de Ci^enciasda Universidade do Porto para obten¸c~ao do grau de Doutor em Matem´atica Departamento de Matem´atica Faculdade de Ci^enciasda Universidade do Porto 2013 To My Parents Acknowledgments I would like to thank my supervisor Prof. Semyon Yakubovich for all the guidance and support he provided for me during my studies at the University of Porto. I would also like to thank Professors: Bagher Nashvadian-Bakhsh, Abdolhamid Ri- azi, Abdolrasoul Pourabbas, Jos´eFerreira Alves for their teaching and supporting me in academic stuffs. Also I would like to thank Professors: Ana Paula Dias, Jos´eMiguel Urbano, Marc Baboulin, Jos´ePeter Gothen and Augusto Ferreira for the nice courses I had with them. I thank Professors J. C. Lagarias, C. Calderon, J. Stopple, and M. Wolf for useful discussions and sending us some relevant references. My sincere thanks to Profes- sor Jean-Louis Nicolas for careful reading some parts of the manuscript, helpful comments and suggestions which rather improved the presentation of the last chap- ter. Also I sincerely would like to thank Professor Paulo Eduardo Oliveira for his kindly assistances and advices as a coordinator of this PhD program. My thanks goes to all my friends which made a friendly environment, in particular Mohammad Soufi Neyestani for what he did before I came to Portugal until now. My gratitude goes to Funda¸c~aoCalouste Gulbenkian for the financial support dur- ing my PhD. I would like to thank all people who taught me something which are useful in my life but I have not mentioned their names. I thank to the members of the juries of my thesis especially Jos´eAssis Azevedo and Ant´onioMachiavelo for their careful reading the thesis and their useful comments. Last but no means least, I would like to thank my family, for their love and en- couragement. ii The scientific work was carried out in 2009-2012 at Department of Mathematics - University of Porto Supervisor Semyon Yakubovich Resumo Durante um estudo da fun¸c~aozeta de Riemann, analizando alguns gr´aficoscom ela rela¸cionadise procurando rela¸c~oesentre a hip´otesede Riemann e o tamanho (valor absoluto) da fun¸c~ao-zeta,observamos uma rela¸c~aointeressante entre esses valores, nomeadamente, que na faixa 0 < σ < 1=2 com jtj ≥ 6:5, se tem (onde, como ´eusual, s = σ + it): jζ(1 − s)j ≤ jζ(s)j Mais tarde, apercebemo-nos que este resultado tinha j´asido demonstrado por Dixon-Schoenfeld e Spira na d´ecada de 1960. No entanto, a nossa demonstra¸c~ao´e diferente e tem a vantagem de envolver, em vez da f´ormula assimpt´oticade Stirling, algumas desigualdades relacionadas com um produto infinito de π e a fun¸c~aoΓ de Euler. O resultado principal do primeiro cap´ıtulo´e,pois, que 1 jζ(1 − s)j ≤ jζ(s)j; for 0 < σ < ; 2 onde a igualdade ocorre s´ose ζ(s) = 0. No segundo cap´ıtulod´a-seum refinamento de estimativas de algumas fun¸c~oes relacionadas com a distribui¸c~aodos n´umerosprimos, tais como as fun¸c~oes e # de Chebyshev, usando uma nova regi~aolivre de zeros e o c´alculode novos zeros da fun¸c~aozeta, obtidas por Kadiri e Gourdon, respectivamente. No terceiro cap´ıtulointroduzimos e investigamos algumas sequ^enciasrelacionadas com um teorema de Robin, que afirma que, a hip´otesede Riemann ´eequivalente `adesigualdade σ(n) < eγn log log n para todos n > 5040, onde σ(n) ´ea soma dos divisores de n e γ ´ea constante de Euler. Com base nesta desigualdade, introduz- imos uma sequ^enciade n´umeros,que apelidamos de extremamente abundantes, e mostramos que a hip´otesede Riemann ´everdadeira se e s´ose existe uma infinidade destes n´umeros.Investigamos ainda algumas das suas propriedades e a estrutura dos n´umerosextremamente abundantes, assim como algumas propriedades dos os n´umerossuperabundantes e colossalmente abundantes. Finalmente apresentamos dois outros conjuntos de n´umeros,relacionados com os n´umerosextremamente abundantes, que mostram o quanto subtil ´ea hip´otesede Riemann. iv Abstract While studying the Riemann zeta-function, observing some graphs related to it and looking for some relation between the Riemann hypothesis and the absolute value of the Riemann zeta function, we noted an interesting relationship between those values, namely, that in the strip 0 < σ < 1=2 with jtj ≥ 6:5, one has (as usual s = σ + it): jζ(1 − s)j ≤ jζ(s)j Later we found that this result had been proved by Dixon-Schoenfeld and Spira in 1960's. Nevertheless, our proof was different and has the advantages of involving, instead of Stirling's asymptotic formula, some inequalities related to an infinite product for π and Euler's Γ-function. The main result of the first chapter is, thus, that 1 jζ(1 − s)j ≤ jζ(s)j; for 0 < σ < ; (0.1) 2 where the equality takes place only if ζ(s) = 0. In the second chapter, we give an improvement for estimates of some functions related to the distribution of primes, such as Chebyshev's and # functions, using some new zero-free region and computations of new zeros of the zeta-function, obtained by Kadiri and Gourdon respectively. In the third chapter, we introduce and investigate some sequences related to Robin's theorem, which states that, the Riemann hypothesis is equivalent to the inequality σ(n) < eγn log log n for all n > 5040, where γ is Euler's constant. Inspired by this inequality, we introduce a sequence of numbers, that we call ex- tremely abundant, and show that the Riemann hypothesis is true if and only if there are infinitely many of these numbers. Moreover, we investigate some of their properties and structure, as well as some properties of superabundant and colos- sally abundant numbers. Finally we introduce two other sets of numbers, related to extremely abundant numbers, that show how subtle the Riemann hypothesis is. v Contents 0 Introduction 1 0.1 A Note About the Riemann Hypothesis . .1 0.2 Equivalent Statements to RH . .4 0.3 Brief Description of the Chapters . .5 0.4 Own papers . .7 1 Riemann Zeta-Function, Its size in Critical Strip and the RH 9 1.1 Preliminaries About the Riemann Zeta Function . 10 1.1.1 Complex-Valued Functions . 10 1.1.2 Representations of Entire Functions . 10 1.1.3 Gamma Function . 11 1.1.4 The Riemann Zeta Function . 12 1.1.5 The Functional Equation . 13 1.1.6 Values of ζ(s) in the Integers . 13 1.1.7 Zeros of the Riemann Zeta Function . 14 1.1.8 Bounds of ζ(s)......................... 16 1.2 On an Inequality for the Riemann Zeta Function in the Critical Strip 16 1.2.1 Auxiliary Lemmas . 17 1.2.2 Proof of the Main Result . 19 1.2.3 An Application to the RH . 23 2 Chebyshev's Functions, Improved Bounds and the RH 26 2.1 Introduction and Preliminary Results . 26 2.1.1 Explicit Bounds for Distribution of Primes . 29 2.1.2 Irregularities of the Distribution of Primes . 33 2.2 A Sufficient Condition for the RH . 34 2.3 Improved Explicit Bounds for Chebyshev's Functions . 35 2.3.1 Improved Explicit Bounds for Large Values of x ....... 35 vi 2.3.2 Improved Explicit Bounds for Moderate Values of x ..... 38 2.4 Improved Explicit Bounds and Distribution of Primes . 39 Y 2.5 Explicit Estimates for (1 + 1=p).................. 42 p≤x 3 Extremely Abundant Numbers and the RH 47 3.1 Introduction and Background . 47 3.2 Extremely Abundant Numbers: Definition and Motivations . 50 3.3 Auxiliary Lemmas . 53 3.4 Some Properties of SA, CA and XA Numbers . 56 3.4.1 SA Numbers . 57 3.4.2 CA Numbers . 68 3.4.3 XA Numbers . 71 3.5 Delicacy of the RH . 75 3.6 Numerical Experiments . 77 vii List of Tables 1 Numerical verification of zeros of the zeta function ([6], p. 39) . .2 2.1 Prime counting function and logarithmic integral[23, p. 2] . 27 Qk 3.1 First 10 extremely abundant numbers (pk] = j=1 pj is primorial of p)................................... 79 ∗ b 3.2 j (x) − xj < x"0; (x ≥ e ), for Theorem 2.20 . 85 ∗ 3.3 ηk for the case j (x) − xj < x"0 .................... 86 ∗ 3.4 ηk for the case j (x) − xj < x"0 .................... 87 ∗ 3.5 ηk for the case j (x) − xj < x"0 .................... 88 viii Notations f(x) = Ω+(g(x)) f(x) > Cg(x) for a suitable constant C > 0 and a sequence x = xn s.t. xn ! 1 f(x) = Ω−(g(x)) f(x) < −Cg(x) for a suitable constant C > 0 and a sequence x = xn s.t. xn ! 1 f(x) = Ω±(g(x)) Both f(x) = Ω+(g(x)) and f(x) = Ω−(g(x)) hold f(x) = Ω(g(x)) jf(x)j = Ω+(g(x)) f(x) = O(g(x)) jf(x)j ≤ Ajg(x)j for some constant A and all values of x > x0 for some x0 f(x) f(x) = o(g(x)) limx!1 g(x) = 0 f(x) f ∼ g limx!1 g(x) = 1 π(x) the number of prime numbers p ≤ x Π(x) Riemann prime counting function; i.e. P 1 1 1=2 1 1=3 pm≤x m = π(x) + 2 π(x ) + 3 π(x ) + ··· Z 1−" dt Z x dt li(x) lim + "!0 0 log t 1+" log t R x dt Li(x) 2 log t ix bxc the integral part of x σ(n) sum of divisors of a positive integer n d(n) number of divisors of a positive integer n φ(n) Euler's totient function: the number of positive integers not exceeding n which are relatively prime to n P #(x) Chebyshev's first function: p≤x log p P (x) Chebyshev's second function: pm≤x log p Q Ψ(n) Dedekind's arithmetical function: n pjn(1 + 1=p) SA Set of superabundant numbers, also abbreviation for superabundant CA Set of colossally abundant numbers, also abbreviation for colossally abundant XA Set of extremely abundant numbers, also abbreviation for extremely abundant x Chapter 0 Introduction Contents 0.1 A Note About the Riemann Hypothesis .