Explicit estimates on prime numbers
This submission has been withdrawn by arXiv admins due to fraudulent affiliation claims by the original submitter.
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arXiv subjects
Yuanyou Cheng
Publications and source records attributed to Yuanyou Cheng.
Further Estimates with Pseudogamma functions
The pseudo-Gamma function is a key tool introduced recently by Cheng and Albeverio in the proof of \break the density hypothesis. This function is doubly symmetric, which means that it is reflectively symmetric about the real axis by the Schwarz principle, whereas it is also reflectively symmmetric about the half line where the real part of the variable is equal to $\tfrac{1}{2}$. In this article, we sharpen the estimate given in the proof of the density hypothesis for this doubly symmetric pseudo-Gamma function on the real axis near the symmetry center by taking a different approach from the way used in the density hypothesis proof directly from the definition, reducing the error caused by the fact that the difference of two pivotal parameters in the definition of the pseudo-Gamma function is much larger than the difference of the variables in this particular case.
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Analytic implication from the prime number theorem
Let $x\ge 2$. The $\psi$-form of the prime number theorem is $\psi(x) =\sum\sb{n \le x}\Lambda(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$, where $H(x)$ is a certain function of $x$ with $0< H(x) \le \tfrac{1}{2}$. Tur\'an proved in 1950 that this $\psi$-form implies that there are no zeros of $\zeta(s)$ for $\Re(s) > h(t)$, where $t=\Im(s)$, and $h(t)$ is a function related to $H(x)$ with $0< h(t) \le \tfrac{1}{2}$, but both $H(x)$ and $h(t)$ are very close to 1. We prove results similar to Tur\'an's, with $H(x)$ and $ h(t)$ in some altered forms without the restriction that $H(x)$ and $h(t)$ are close to 1. The proof involves slightly revising and applying Tur\'an's power sum method and using the Lindel\"of hypothesis in the zero growth rate form, which is proved recently.
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Proof of the strong Linderlof hypothesis
The Riemann zeta-function $\zeta(s)$ is a meromorphic complex-valued function of the complex variable $s$ with the unique pole at $s=1$. It plays a central role in the studies of prime numbers. The upper bound in the critical strip $0\le \Re(s) \le 1$ is an important element in this study. The Lindel\"of hypothesis conjectured in 1908 asserts that $|\zeta(\tfrac{1}{2} +it)| =O(t\sp{\epsilon})$ for sufficiently large $t$. In 1921, Littlewood showed that this is equivalent to an estimate on the number of zeros in certain regions. We use the pseudo-Gamma function recently devised by Cheng and Albeverio in proving the density hypothesis to validate an estimate on the growth rate of zeros and obtain a slightly sharper result than the one which is equivalent with the Lindel\"of hypothesis. Thus, in particular, we have a proof of the Lindel\"of hypothesis.
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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 $\varpi(x) =O(x\sp{1/2} \log\sp{2} x)$, where $\varpi(x) =\sum\sb{n\le x}\ \bigl(\Lambda(n) -1\big)$ with the sum running through the set of all natural integers. Let ${\mathsf Z}(s) = -\tfrac{\zeta\sp{\prime}(s)}{\zeta(s)} -\zeta(s)$. We use the classical integral formula for the Heaviside function in the form of ${\mathsf H}(x) =\int\sb{m -i\infty} \sp{m +i\infty} \tfrac{x\sp{s}}{s} \dd s$ where $m >0$, and ${\mathsf H}(x)$ is 0 when $\tfrac{1}{2} 1$. However, we diverge from the literature by applying Cauchy's residue theorem to the function ${\mathsf Z}(s) \cdot \tfrac{x\sp{s}} {s}$, rather than $-\tfrac{\zeta\sp{\prime}(s)} {\zeta(s)} \cdot \tfrac{x\sp{s}}{s}$, 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 $\varpi(x) =O(x\sp{\theta} \log\sp{2} x)$, 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 Lindel\"of 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 Tur\'an.
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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 $N(\lambda, T) =O\bigl(T\sp{2(1-\lambda) +\epsilon} \bigr)$ for any $\epsilon >0$, where $N(\lambda, T)$ is the number of zeros of $\zeta(s)$ when $\Re(s) \ge\lambda$ 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 $T\sp{\epsilon}$ being replaced by $\log T$. In an attempt to transform Backlund's 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 $T\ge 2445999554999$.
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