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Register a new userOn the zeros of the Riemann zeta function
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Authors
Jorma Jormakka
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In 2008 I thought I found a proof of the Riemann Hypothesis, but there was an error. In the Spring 2020 I believed to have fixed the error, but it cannot be fixed. I describe here where the error was. It took me several days to find the error in a careful checking before a possible submission to a payable review offered by one leading journal. There were three simple lemmas and one simple theorem, all were correct, yet there was an error: what Lemma 2 proved was not exactly what Lemma 3 needed. So, it was the connection of the lemmas. This paper came out empty, but I have found a different proof of the Riemann Hypothesis and it seems so far correct. In the discussion at the end of this paper I raise a matter that I think is of importance to the review process in mathematics.
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We improve the estimation of the distribution of the nontrivial zeros of Riemann zeta function $zeta(sigma+it)$ for sufficiently large $t$, which is based on an exact calculation of some special logarithmic integrals of nonvanishing $zeta(sigma+it)$ along well-chosen contours. A special and single-valued coordinate transformation $s=tau(z)$ is chosen as the inverse of $z=chi(s)$, and the functional equation $zeta(s) = chi(s)zeta(1-s)$ is simplified as $G(z) = z, G_-(frac{1}{z})$ in the $z$ coordinate, where $G(z)=zeta(s)=zetacirctau(z)$ and $G_-$ is the conjugated branch of $G$. Two types of special and symmetric contours $partial D_{epsilon}^1$ and $partial D_{epsilon}^2$ in the $s$ coordinate are specified, and improper logarithmic integrals of nonvanishing $zeta(s)$ along $partial D_{epsilon}^1$ and $partial D_{epsilon}^2$ can be calculated as $2pi i$ and $0$ respectively, depending on the total increase in the argument of $z=chi(s)$. Any domains in the critical strip for sufficiently large $t$ can be covered by the domains $D_{epsilon}^1$ or $D_{epsilon}^2$, and the distribution of nontrivial zeros of $zeta(s)$ is revealed in the end, which is more subtle than Riemanns initial hypothesis and in rhythm with the argument of $chi(frac{1}{2}+it)$.
The Hardy hypothesis, as an analogue to the Riemann hypothesis for the Riemann zeta function, is a conjecture proposed by Hardy in 1940, that all of the nontrivial zeros for the Ramanujan zeta function have a real part equal to 6. In this paper, we propose the power series expansion for the entire Ramanujan zeta function using the work of Mordell. Then, we suggest an alternative infinite product for the entire Ramanujan zeta function derived from the work of Conrey and Ghosh. We also establish the class of the entire Ramanujan zeta function related to the functional equation coming from Wilton. Motivated by the work of Lekkerkerker, we prove an conjecture due to Bruijn that all of the zeros of the Ramanujan Xi function are nonzero real numbers. From theory of the entire functions, we also prove that the Hardy hypothesis is true.
We study the limit distribution of eigenvalues of a Ruelle operator (which is also called the Thurston pushforward operator) for the dynamical system $z mapsto z^2+c$ when $c<-2$ and tends to $-2$.
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $zeta(s)$, $s=sigma+i t$, $0leq sigma leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $Phi(u,v,beta)$, $uin mathbb{C}$, $vin mathbb{C}$, $beta in mathbb{R}$. Generalizing the methodology used in the study of $zeta(s)$, we derive asymptotic formulae for $Phi(u,v,beta)$.
This is a review of some of the interesting properties of the Riemann Zeta Function.
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