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Register a new userAverage Orders of the Euler Phi Function, The Dedekind Psi Function, The Sum of Divisors Function, And The Largest Integer Function
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N. A. Carella
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Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ varphi(n)$ be the Euler totient function. The result $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+Oleft ( x(log x)^{2/3}(loglog x)^{1/3}right ) $ was proved very recently. This note presents a short elementary proof, and sharpen the error term to $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+O(x) $. In addition, the first proofs of the asymptotics formulas for the finite sums $ sum_{nleq x}psi([x/n])=(15/pi^2)xlog x+O(xlog log x) $, and $ sum_{nleq x}sigma([x/n])=(pi^2/6)xlog x+O(x log log x) $ are also evaluated here.
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Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ sum_{pleq x}sigma([x/p])=c_0xlog log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ sum_{pleq x}sigma([x/(p+a)])=c_0xlog log x+O(x) $ for any fixed integer $a$.
We obtain reasonably tight upper and lower bounds on the sum $sum_{n leqslant x} varphi left( leftlfloor{x/n}rightrfloorright)$, involving the Euler functions $varphi$ and the integer parts $leftlfloor{x/n}rightrfloor$ of the reciprocals of integers.
Let $eta$ be the weight $1/2$ Dedekind function. A unification and generalization of the integrals $int_0^infty f(x)eta^n(ix)dx$, $n=1,3$, of Glasser cite{glasser2009} is presented. Simple integral inequalities as well as some $n=2$, $4$, $6$, $8$, $9$, and $14$ examples are also given. A prominent result is that $$int_0^infty eta^6 (ix)dx= int_0^infty xeta^6 (ix)dx ={1 over {8pi}}left({{Gamma(1/4)} over {Gamma(3/4)}}right)^2,$$ where $Gamma$ is the Gamma function. The integral $int_0^1 x^{-1} ln x ~eta(ix)dx$ is evaluated in terms of a reducible difference of pairs of the first Stieltjes constant $gamma_1(a)$.
In this paper, some sufficient conditions for the differentiability of the $n$-variable real-valued function are obtained, which are given based on the differentiability of the $n-1$-variable real-valued function and are weaker than classical conditions.
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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