Spectral Analysis of Arithmetic Functions
Keywords:
Prime gaps, Lambert series, Number theory, Arithmetic functionsAbstract
This paper explores the normalization of Fourier series kernels associated with elementary arithmetic functions. By analyzing key functions such as divisor related functions, the prime gap function and the inverse prime counting function. With these results, a framework for their spectral decomposition in terms of complex exponential functions was developed. Through contour integration techniques and normalization of the Dirac delta function, new methods can be derived from the analytic structure of these functions. These findings contribute to the broader understanding of arithmetic functions and their link to Fourier series.
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