Related papers: Biases in prime factorizations and Liouville funct…
In this paper, we introduced the theory of the sieve function transformation. Using the principle of sieve function transformation, we improved sieve method, and obtained the difference range of similar sieve function values. For this, we…
For two odd primes $p$ and $q$ such that $p<q$, let $A(p,q):=(a_k)_{k=1}^{\infty}$ be the arithmetic progression whose $k$th term is given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here we conjecture that for every positive…
We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
We give new characterizations of the Midy's property and using these results we obtain a new proof of a special case of the Dirichlet's theorem about primes in arithmetic progression.
We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-\epsilon}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and…
In this article I present results from a statistical study of prime numbers that shows a behaviour that is not compatible with the thesis that they are distributed randomly. The analysis is based on studying two arithmetical progressions…
Polignac [1] conjectured that for every even natural number $2k (k\geq1)$, there exist infinitely many consecutive primes $p_n$ and $p_{n+1}$ such that $p_{n+1}-p_n=2k$. A weakened form of this conjecture states that for every $k\geq1$,…
{\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the…
We use a method of rotations to study the $L^p$ boundedness, $1<p<\infty$, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric $\alpha$-stable processes, $0<\alpha<2$. Our proof does not…
We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Ap\'ery-like recurrence relation: these include Ap\'ery, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and…
We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann…
We conjecture a set of differential equations that characterizes the Liouville irregular states of half-integer ranks, which extends the generalized AGT correspondence to all the $(A_1,A_\text{even})$ and $(A_1,D_\text{odd})$ types…
The Bateman--Horn Conjecture predicts how often an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes prime values. We demonstrate that with sufficient averaging in the coefficients of $f$ (viz. exponential in the size of the inputs),…
Assuming a uniform $q$-variant of the prime $k$-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus $q$ as the residue classes vary. Consequently, depending on the size of $\varphi(q)$,…
For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x,…
The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature.…
As for the Fourier transforms of positive and integrable functions supported in the unit interval, we make a list of improvements for P\'olya's results on the distribution of their positive zeros and give new sufficient conditions under…
We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight…
For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for…