Related papers: Result on the Mobius Function over Shifted Primes

In the present paper we obtain several new results related to the problem of upper bound estimates for the number of solutions of the congruence $$ x^{x}\equiv \lambda\pmod p;\quad x\in \mathbb{N},\quad x\le p-1, $$ where $p$ is a large…

We apply upper and lower compensated convex transforms, which are `tight' one-sided approximations of a given function, to the extraction of fine geometric singularities from semiconvex/semiconcave functions and DC-functions in…

In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We…

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…

We consider a wide class of summatory functions F{f;N,p^m}=\sum_{k\leq N}f(p^m k), m\in \mathbb Z_+\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\in \mathbb Z_+, where p is a prime number.…

New asymptotic relations between the $L_p$-errors of best approximation of univariate functions by algebraic polynomials and entire functions of exponential type are obtained for $p\in (0,\iy]$. General asymptotic relations are applied to…

Let $A_f(1,n)$ be the normalized Fourier coefficients of a Hecke-Maass cusp form $f$ for $SL_3(\mathbb{Z})$ and $$ r_3(n)=\#\left\{(n_1,n_2,n_3)\in \mathbb{Z}^3:n_1^2+n_2^2+n_3^2=n\right\}. $$ Let $1\leq h\leq X$ and $\phi(x)$ be a smooth…

Let $p_n$ denote the $n$-th prime number, and let $d_n=p_{n+1}-p_{n}$. Under the Hardy--Littlewood prime-pair conjecture, we prove \begin{align*} \sum_{n\le X}\frac{\log^{\alpha}d_n}{d_n} \sim\begin{cases} \frac{X\log\log\log X}{\log…

Asymptotic expansions are derived for associated Legendre functions of degree $\nu$ and order $\mu$, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument…

We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results…

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$…

The cumulative sum (CUSUM) process is often used in change point analysis to detect changes in the mean of sequentially observed data. We provide a full description of the asymptotic distribution of $L^p, 1\leq p <\infty$, functionals of…

Let $\lambda$ be an uncountable cardinal such that $2^{< \lambda } = \lambda$. Working in the setup of generalized descriptive set theory, we study the structure of $\lambda^+$-Borel measurable functions with respect to various kinds of…

Completely multiplicative functions whose sum is zero ($CMO$).The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all…

We give explicit numerical values with 100 decimal digits for the Mertens constant involved in the asymptotic formula for $\sum\limits_{\substack{p\leq x p\equiv a \bmod{q}}}1/p$ and, as a by-product, for the Meissel-Mertens constant…

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on ${\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums, we…

We extend the Matom\"{a}ki-Radziwi\l\l{} theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a…

Let $G=(V,E)$ be a finite connected weighted graph, and assume $1\leq\alpha\leq p\leq q$. In this paper, we consider the following $p$-th Yamabe type equation $$-\Delta_pu+hu^{q-1}=\lambda fu^{\alpha-1}.$$ on $G$, where $\Delta_p$ is the…

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{\deg f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by…

It is shown that that for every Darboux function $F$ there is a non-constant continuous function $f$ such that $F+f$ is still Darboux. It is shown to be consistent --- the model used is iterated Sacks forcing --- that for every Darboux…