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- W2425119069 abstract "A form of the exponential Mangoldt function is derived using indicator functions. The function's relationship to other important number theoretic functions are derived and discussed. The Mangoldt function, is defined as . So, by the property of the natural logarithm, the exponential Mangoldt function is defined as The function can be represented as a sum using indicator functions as I shall prove in the following theorem. Theorem: . Proof: The two expressions within parenthesis that are a difference of floor functions are indicator functions of the form . We can use this form to define the first indicator function in the sum, If we define to be and to be , then we have Wilson's theorem states . Using this we have which is the characteristic function of the primes . For the other indicator function in the sum , note that the exponents in the prime factorization of are going to be larger than the exponents of the prime factorization of , because is raised to the power of So, only the prime factors of and matter in whether or not Considering the factors, can only divide if is a multiple of the product of the distinct prime factors of , which is defined as From this we have . Thus, for the two indicator functions multiplied together in the sum we have Summing this from to we have . So, if we multiply the argument of this sum by we have This is just the exponential Mangoldt function with value instead of at non-prime powers. Thus, adding the characteristic function of non-prime powers will give the exponential Mangoldt function. But, this is just Thus, using the two sums directly above we have" @default.
- W2425119069 created "2016-06-24" @default.
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- W2425119069 date "2016-06-01" @default.
- W2425119069 modified "2023-09-23" @default.
- W2425119069 title "On The Exponential Mangoldt Function" @default.
- W2425119069 hasPublicationYear "2016" @default.
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