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MOULOUD GOUBI UMMTO University Department of Mathematics 15000 Tizi-Ouzou ALGERIA Algebra and Number Theory Laboratory USTHB Algiers mouloud.goubi@ummto.dz Abstract: In this work we are interested by cotangent sum related to Estermann zeta function in rational arguments. In the first place we look at the maximum and the moment as they did H. Maier and M. Th. Rassias in short interval and get some interesting new identities. Afterwards, based on some recent results, we study special cases where we provide the series expansion of this sum. We end the work by the associated arithmetical problem; which consists to find the relation between an integer and its inverse in congruence theory. Key–Words: Cotangent sums; Estermanns zeta function; Riemann zeta function; Riemann Hypothesis. Received: January 27, 2020. Revised: March 17, 2020. Accepted: March 23, 2020. Published: March 31, 2020.

On the Cotangent Sums Related to Estermann Zeta Function and Arithmetic Properties of their Arguments