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Register a new userSome Interesting Properties of the Riemann Zeta Function
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Authors
Johar M. Ashfaque
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This is a review of some of the interesting properties of the Riemann Zeta Function.
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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.
We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $zeta(s)$. In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where $S(t)$ denotes the argument of $zeta(s)$ on the critical line and $h ll 1 / log T$. We also use these moments to prove explicit results on the density of the nontrivial zeroes of $zeta(s)$ of a given multiplicity.
Voronins theorem on the `Universality of Riemann zeta function is shown to imply that Riemann zeta function is a fractal (in the sense that Mandelbrot set is a fractal) and a concrete ``representation of the ``giant book of theorems that Paul Halmos referred to.
We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - epsilon}$
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 $Phi(u,v,beta)$.
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