Robin inequality for n/phi(n)

Petites valeurs de la fonction d'Euler

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the product of the first $k$ primes. In this article, we consider the function $c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$. Under Riemann's hypothesis, it is proved that $c(N_k)$ is bounded and explicit bounds are given while, if Riemann's hypothesis fails, $c(N_k)$ is not bounded above or below.

Preface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity 325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.

Let Λ(n) be the von Mangoldt function and τ(n) the divisor function. We focus on the summation T ( x ) = ∑ 1 < n ≤ x Λ ( n ) τ ( n − 1 ) . Under the Riemann hypothesis related to Dirichlet L-functions, it is proved that T ( x ) = x P 1 ( log x ) + O ( x 1 − 𝜃 ) holds for 𝜃<16×10−4. Here P1(t) is a polynomial of degree 1.

Let ψ(n)be Euler function,σ(n)denote the sum of divisors of n and d(n)be the divisor function.In this paper we prove the following four theorems.Theorem 1.For any given sequence of κ non-negative numbers α_1,…,α_κand e0,there exists a prime number p such that(?) (1)There exist positive constants с-0(α,e)and Х_0(α,e)such that the numberof primes р satisfying(1)in the interval 1р≤x is greater than(?)Theorem 2 is obtained from Theorem 1 by replacing the letters ψ by σ,с_0 by с-1 and Х_0 by Х_1.Theorem 3.For any given natural number к,there exists a constant γdepending on κ only such that for any given sequence of κ+1 positive integersα_0,α-1,…,α_κ,there exists a prime number р,such that (?) (2)There exist positive constant с_2(α)and Х_2(α)such that the number ofprimes р satisfying(2)in the interval 1≤р≤x is greater than(?)Theorem 4.For any given sequence of κ numbers α_1,…,α_κ,where α_i=0or +∞(1≤i≤κ),there exists an infinite sequence of prime numbers{р_j}(j=1,2,…)such that(?)These theorems improve some results of the author,Schinzel,andShao.The proof of these theorems depends on the followingFundamental Lemma.Let(?)be given natural numbers,where q_(μυ)(0≤μ≤κ,1≤υ≤t_μ)are prime num-bers greater than κ+1 and relatively prime in pairs.If xZ(m_0m_1…m_κ)~2,and N_z(x)denote the number of positive solu-tions(р,x_0,x_1…,x_κ)of the system of equations(?)satisfying the conditions(?)where р and р′denote primes,than there exist positive constants с_3,Х_3 depen-dent only on m_i,and α,dependent only on κ such that(?)If xZ(m_0…m_κ)~2,λ is a given positive number in the interval 1≤λ≤≤(m_0…m_κ)~2 satisfying(λ,m_0…m_κ)=1 and р_1р_2…р_r≤Z are all primenumbers that do not divide m_0…m_κ and do not exceed Z,and if α_(ij)(1≤i≤≤r,1≤j≤κ+1)are given positive numbers satisfying the conditions 1≤α_(ij)р_i and j_≠j_2 impries α_(ij_1)≠α_(ij_2),then we can define M_z(x)as the numberof primes р satisfying the conditions1р≤x,р≡λ(mod(m_0…m_κ)~2),р(?)α_(ij)(mod р_i)(1≤i≤r,1≤j≤κ+1).Fundamental Lemma obviously follows from the following two lemmas.Lemma 1.There exist α_(ij) and λ such that M_z(x)≤N_z(x).Lemma 2.There exist positive constants с_4,Х_4,dependent only on m_i,and positive constant β,dependent only on κ,such that(?)for any given λ and α_(ij).The proof of Lemma 2 depends essentially on the methods of Линник andRe(?)yi.

Let sigma (n) denote the sum of divisors function of a positive integer n. Robin proved that the Riemann hypothesis is true if and only if the inequality sigma (n) < textrm{e}^{gamma }n log log n holds for every integer n > 5040, where gamma is the Euler–Mascheroni constant. In this paper we establish a new family of integers for which Robin’s inequality sigma (n) < textrm{e}^{gamma }n log log n hold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev’s vartheta -function and for some product defined over prime numbers.

This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.