Related papers: A Dedekind Psi Function Inequality

For a positive integer k and an arbitrary integer h, the Dedekind sum s(h,k) was first studied by Dedekind because of the prominent role it plays in the transformation theory of the Dedekind eta-function, which is a modular form of weight…

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid \phi(n)$, where $\phi$ is Euler's totient function. We show that the normal order of pr$(n)$ is $\log\log…

For $k\geq 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function \[f_k(n)=\frac{\log d_k(n)\log\log n}{\log k\log n},\quad n\geq 3\] which has a maximum value $\lambda(k)$ at a…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

Assume that $N$ is a sufficiently large positive number. In this paper we show that for a small constant $\varepsilon>0$, the logarithmic inequality \begin{equation*} \big|p_1\log p_1+p_2\log p_2+p_3\log p_3-N\big|<\varepsilon…

For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\le C_2 e^{-\gamma} \phi(2n_k)/\log\log(2n_k)$ holds for all…

We study non-vanishing of Dirichlet $L$-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if $\psi$ is a real primitive character modulo $D \in…

Let $s(m,n)$ denote the classical \DED sum, where $n$ is a positive integer and $m\in\{0,1,\ldots, n-1\}$, $(m,n)=1$. For a given positive integer $k$, we describe a set of at most $k^2$ numbers $m$ for which $s(m,n)$ may be $\ge s(k,n)$,…

Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and…

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33…

We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-\gamma})^{-1} = 2.280...$ and $\gamma$ is…

If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{\sigma(p^k)}{p^k}$ and $\frac{\sigma(m^2)}{m^2}$, and the ratios…

Let $\psi_{\mathbb K}$ be the Chebyshev function of a number field $\mathbb K$. Let $\psi^{(1)}_{\mathbb K}(x):=\int_{0}^{x}\psi_{\mathbb K}(t)\,d t$ and $\psi^{(2)}_{\mathbb K}(x):=2\int_{0}^{x}\psi^{(1)}_{\mathbb K}(t)\,d t$. We prove…

For $n\in \mathbb{N}$ let $S_n$ be the smallest number $S>0$ satisfying the inequality $$ \int_K f \le S \cdot |K|^{\frac 1n} \cdot \max_{\xi\in S^{n-1}} \int_{K\cap \xi^\bot} f $$ for all centrally-symmetric convex bodies $K$ in…

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^\gamma <\infty$, $\gamma>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k…

Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions,…

We study the asymptotic behaviour of the classical Dedekind sums $s(s_k/t_k)$ for the sequence of convergents $s_k/t_k$ $k\ge 0$, of the transcendental number \BD \sum_{j=0}^\infty\frac {1}{b^{2^j}},\ b\ge 3. \ED In particular, we show that…

For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is…

We study the value distribution of the Sudler product $P_N(\alpha) := \prod_{n=1}^{N}\lvert2\sin(\pi n \alpha)\rvert$ for Lebesgue-almost every irrational $\alpha$. We show that for every non-decreasing function $\psi: (0,\infty) \to…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with…