Proof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method

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.

Proof of the strong Density Hypothesis

The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that the non-trivial zeros of $zeta(s)$ lie on the line $Re(s) =1/2$. The density hypothesis is a conjectured estimate $N(lambda, T) =Obigl(Tsp{2(1-lambda) +epsilon} bigr)$ for any $epsilon >0$, where $N(lambda, T)$ is the number of zeros of $zeta(s)$ when $Re(s) gelambda$ and $0 <Im(s) le T$, with $1/2 le lambda le 1$ and $T >0$. The Riemann-von Mangoldt Theorem confirms this estimate when $lambda =1/2$, with $Tsp{epsilon}$ being replaced by $log T$. In an attempt to transform Backlunds proof of the Riemann-von Mangoldt Theorem to a proof of the density hypothesis by convexity, we discovered a different approach utilizing an auxiliary function. The crucial point is that this function should be devised to be symmetric with respect to $Re(s) =1/2$ and about the size of the Euler Gamma function on the right hand side of the line $Re(s) =1/2$. Moreover, it should be analytic and without any zeros in the concerned region. We indeed found such a function, which we call pseudo-Gamma function. With its help, we are able to establish a proof of the density hypothesis. Actually, we give the result explicitly and our result is even stronger than the original density hypothesis, namely it yields $N(lambda, T) le 8.734 log T$ for any $1/2 < lambda < 1$ and $Tge 2445999554999$.