7th ISF

MATH016 - New Inequalities Between Arithmetic Functions

In this paper, we introduce a relationship between arithmetic functions, especially phi function and sigma function from the investigation of their asymptotic behaviour as the input size grows. Moreover, we use the properties of the prime numbers and multiplicative functions to demonstrate how these tools can be used to establish new inequalities such as (σ(n))/n≤3/4 (1+1/P(n/2) ) √(d(n) ),∀n≥85 where P(k) a denotes the least prime not more than or equal to k. Finally, we provide numerical evidence to support our theoretical result

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