On an Approximation of Divisor Sum Functions With Bernoulli Polynomials and the Hardy Littlewood Function

Overview

Is a Bessel representation of exponential sums corresponding to divisors obtainable, and if so, viable? In an original expression I derived relating the divisor functions at individual points and their averages by exploiting the periodicity of roots of unity, in a sense inverting it, there is an exponential sum called the Hardy Littlewood function. Moreover, Voronoi uses a generating function transform using complex analytical tools to describe the error between the average of the divisor function and its asymptotic main terms. With both in mind, I similarly used a complex analytical generating function transform. To get an exact representation of the sum, I changed the contour from a circle to a keyhole. The result of this procedure is a conversion between any exponential sum and a Hankel transform. To fully express the exponential sum in terms of a Hankel Transform, a subset of the geometric series that’s related to the amplitude function of the exponential sum must be studied. Also, I expressed the Hankel Transform in terms of the FHA cycle, and showed a correlation between an oscillatory integral and exponential sum: similar objects with different measures. The literature shows that both of these can be studied: with methods from Mackinnon’s paper on asymptotic expansions of Hankel Transforms and the saddle point method. These results open up a new gap for sums involving subsets of the geometric series by creating context for them, and different methods besides the Van Der Corput processes A and B are discovered.

Awards (3)

• Mu Alpha Theta, National High School and Two-Year College Mathematics Honor Society: Second Award of $1,000 $1,000
• American Mathematical Society: One-Year Membership to American Mathematical Society to each winner (7 winning projects, up to 3 team members per project)
• American Mathematical Society: First Award of $2,000 $2,000

Competition history

• ISEF 2024 Mathematics · Entry MATH018 · Los Angeles, California, United States

Resources

• ISEF abstract page ↗

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