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تسجيل مستخدم جديدProof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method
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Yuanyou Cheng
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The Riemann hypothesis is equivalent to the $varpi$-form of the prime number theorem as $varpi(x) =O(xsp{1/2} logsp{2} x)$, where $varpi(x) =sumsb{nle x} bigl(Lambda(n) -1big)$ with the sum running through the set of all natural integers. Let ${mathsf Z}(s) = -tfrac{zetasp{prime}(s)}{zeta(s)} -zeta(s)$. We use the classical integral formula for the Heaviside function in the form of ${mathsf H}(x) =intsb{m -iinfty} sp{m +iinfty} tfrac{xsp{s}}{s} dd s$ where $m >0$, and ${mathsf H}(x)$ is 0 when $tfrac{1}{2} <x <1$, $tfrac{1}{2}$ when $x=1$, and 1 when $x >1$. However, we diverge from the literature by applying Cauchys residue theorem to the function ${mathsf Z}(s) cdot tfrac{xsp{s}} {s}$, rather than $-tfrac{zetasp{prime}(s)} {zeta(s)} cdot tfrac{xsp{s}}{s}$, so that we may utilize the formula for $tfrac{1}{2}< m <1$, under certain conditions. Starting with the estimate on $varpi(x)$ from the trivial zero-free region $sigma >1$ of ${mathsf Z}(s)$, we use induction to reduce the size of the exponent $theta$ in $varpi(x) =O(xsp{theta} logsp{2} x)$, while we also use induction on $x$ when $theta$ is fixed. We prove that the Riemann hypothesis is valid under the assumptions of the explicit strong density hypothesis and the Lindelof hypothesis recently proven, via a result of the implication on the zero free regions from the remainder terms of the prime number theorem by the power sum method of Turan.
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