Lagarias’ equivalence to the Riemann hypothesis

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SUMMARY

Lagarias’ equivalence to the Riemann hypothesis posits that for n > 1, the inequality hn + ehn ln hn > σn holds, where hn is the n-th harmonic number and σn is the divisor function of n. This conjecture is linked to a $1,000,000 prize for its proof, as detailed on www.claymath.org. However, some participants argue that evaluating the divisor function for all integers n is impractical, and suggest that more promising avenues may arise from the Hilbert-Polya conjecture or the conditions of a Fourier transform with only real zeros.

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  • Understanding of harmonic numbers and their properties
  • Familiarity with the divisor function and its implications in number theory
  • Knowledge of the Riemann hypothesis and its significance in mathematics
  • Basic concepts of Fourier transforms and their applications
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  • Research the Hilbert-Polya conjecture and its relevance to the Riemann hypothesis
  • Study the properties and applications of the divisor function in number theory
  • Explore harmonic numbers and their relationships to other mathematical concepts
  • Learn about Fourier transforms and the conditions for real zeros in mathematical analysis
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Mathematicians, number theorists, and researchers interested in the Riemann hypothesis and its related conjectures, particularly those exploring harmonic numbers and divisor functions.

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Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if
hn := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and
σn := divisor function of n := sum of positive divisors of n, then if n > 1,
hn + ehn ln hn > σn.

There is a $1,000,000 prize for the proof of this at www.claymath.org
 
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For me this approach is a bit of nonsense, since you can not evaluate the divisor function for every n=0,1,2,3,4,... not even an asymptotic formula (with a good remainder) is known for divisor function

I think that the most promising approach will come from Hilbert-Polya conjecture or the condition of a Fourier transform having only real zeros.
 

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