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A variety of Euler's sum of powers conjecture

Tianxin Cai, Yong Zhang (2021)

Czechoslovak Mathematical Journal

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system n = a 1 + a 2 + ⋯ + a s - 1 , a 1 a 2 ⋯ a s - 1 ( a 1 + a 2 + ⋯ + a s - 1 ) = b s has positive integer or rational solutions n , b , a i , i = 1 , 2 , ⋯ , s - 1 , s ≥ 3 . Using the theory of elliptic curves, we prove that it has no positive integer solution for s = 3 , but there are infinitely many positive integers n such that it has a positive integer solution for s ≥ 4 . As a corollary, for s ≥ 4 and any positive integer n , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...

A zero density result for the Riemann zeta function

Habiba Kadiri (2013)

Acta Arithmetica

We prove an explicit bound for N(σ,T), the number of zeros of the Riemann zeta function satisfying ℜ𝔢 s ≥ σ and 0 ≤ ℑ𝔪 s ≤ T. This result provides a significant improvement to Rosser's bound for N(T) when used for estimating prime counting functions.

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