2019arXiv (Cornell University)Open access

Closed-form formula of Riemann zeta function and eta function for all\n non-zero given complex numbers via sums of powers of complex functions to\n disprove Riemann Hypothesis

Dagnachew Jenber Negash

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Abstract

An explicit identity of sums of powers of complex functions presented via\nthis a closed-form formula of Riemann zeta function produced at any given\nnon-zero complex numbers. The closed-form formula showed us Riemann zeta\nfunction has no unique solution for any given non-zero complex numbers which\nmeans Riemann zeta function is entirely divergent. Infinitely many zeros of\nRiemann zeta function produced unfortunately those zeros also gives us non-zero\nvalues of Riemann zeta function. Among those zeros some of them are zeros of\nRiemann hypothesis. The present paper also discussed on eta\nfunction(alternating Riemann zeta function) with exactly the same behavior as\nRiemann zeta function.\n

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An explicit identity of sums of powers of complex functions presented via\nthis a closed-form formula of Riemann zeta function produced at any given\nnon-zero complex numbers. The closed-form formula showed us Riemann zeta\nfunction has no unique solution for any given non-zero complex numbers which\nmeans Riemann zeta function is entirely divergent. Infinitely many zeros of\nRiemann zeta function produced unfortunately those zeros also gives us non-zero\nvalues of Riemann zeta function. Among those zeros some of them are zeros of\nRiemann hypothesis. The present paper also discussed on eta\nfunction(alternating Riemann zeta function) with exactly the same behavior as\nRiemann zeta function.\n

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Available abstract

An explicit identity of sums of powers of complex functions presented via\nthis a closed-form formula of Riemann zeta function produced at any given\nnon-zero complex numbers. The closed-form formula showed us Riemann zeta\nfunction has no unique solution for any given non-zero complex numbers which\nmeans Riemann zeta function is entirely divergent. Infinitely many zeros of\nRiemann zeta function produced unfortunately those zeros also gives us non-zero\nvalues of Riemann zeta function. Among those zeros some of them are zeros of\nRiemann hypothesis. The present paper also discussed on eta\nfunction(alternating Riemann zeta function) with exactly the same behavior as\nRiemann zeta function.\n

Key concepts: Riemann zeta function, Riemann Xi function, Particular values of Riemann zeta function, Riemann hypothesis, Zero (linguistics), Explicit formulae, Arithmetic zeta function, Mathematics

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Closed-form formula of Riemann zeta function and eta function for all\n non-zero given complex numbers via sums of powers of complex functions to\n disprove Riemann Hypothesis — Research Paper | ScholarLens