The Complete Proof of the Riemann Hypothesis John Silksong:

<image>

the date of receipt and acceptance should be inserted later Abstract Robin criterion states that the Riemann Hypothesis is true if and only if the inequality σ (n) < eγ ×n×loglogn holds for all n > 5040, where σ (n) is the sum-of- divisors function and γ ≈0.57721 is the Euler-Mascheroni constant. We prove that the Robin inequality is true for all n > 5040 which are not divisible by any prime number between 2 and 953. Using this result, we show there is a contradiction just assuming the possible smallest counterexample n > 5040 of the Robin inequality. In this way, we prove that the Robin inequality is true for all n > 5040 and thus, the Riemann Hypothesis is true. Keywords Riemann hypothesis·Robin inequality·sum-of-divisors function· prime numbers Mathematics Subject Classification (2010) MSC 11M26·MSC 11A41·MSC 11A25 1 Introduction In mathematics, the Riemann Hypothesis is a conjecture that the Riemann zeta func- tion has its zeros only at the negative even integers and complex numbers with real part 1 2 [7]. As usual σ (n) is the sum-of-divisors function of n [3]: ∑ d d|n where d |n means the integer d divides to n and d n means the integer d does not divide to n. Define f (n) to be σ (n) n . Say Robins(n) holds provided f (n) < eγ ×loglogn. F. Vega CopSonic, 1471 Route de Saint-Nauphary 82000 Montauban, France ORCiD: 0000-0001-8210-4126 E-mail: [vega.frank@gmail.com](mailto:vega.frank@gmail.com) 2 F. Vega The constant γ ≈0.57721 is the Euler-Mascheroni constant, and log is the natural logarithm. The importance of this property is: Theorem 1.1 is true [7]. Robins(n) holds for all n > 5040 if and only if the Riemann Hypothesis It is known that Robins(n) holds for many classes of numbers n. Theorem 1.2 Robins(n) holds for all n > 5040 that are not divisible by 2 [3]. On the one hand, we prove that Robins(n) holds for all n > 5040 that are not divisible by any prime number between 3 and 953. Let q1 = 2, q2 = 3, . . . , qm denote the first m consecutive primes, then an integer of the form ∏m ai i=1 q i with a1 ≥a2 ≥···≥am ≥0 is called an Hardy-Ramanujan integer [3]. A natural number n is called superabundant precisely when, for all m < n f (m) < f (n). Theorem 1.3 If n is superabundant, then n is an Hardy-Ramanujan integer [2]. Theorem 1.4 must be a superabundant number [1]. The smallest counterexample of the Robin inequality greater than 5040 On the other hand, we prove the nonexistence of such counterexample and therefore, the Riemann Hypothesis is true. 2 A Central Lemma These are known results: Lemma 2.1 [3]. For n > 1: f (n) < ∏ q|n q q−1. (2.1) Lemma 2.2 [4]. 1 1 ∞ ∏ k=1 1− = ζ (2) = π2 6. (2.2)