## Number Theory   ## Proof of Lagarias’s Elementary Version of the Riemann Hypothesis

Authors: Stephen Marshall

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zero’s only at the negative even integers and complex numbers with real part 1/2 The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). It was proposed by Bernhard Riemann (1859), after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies significant results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series: ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ... The Riemann hypothesis asserts that all interesting solutions of the equation: ζ(s) = 0 lie on a certain vertical straight line. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Hn = 1 + 1/2+1/3+1/4+⋯+ 1/n = ∑_(n=1)^n▒1/n Harmonic numbers have been studied since early times and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.

### Submission history

[v1] 2018-12-04 16:03:11
[v2] 2018-12-06 08:38:16
[v3] 2018-12-10 15:13:28