arXiv subjects

Zarullo Rakhmonov

Publications and source records attributed to Zarullo Rakhmonov.

3 recordsLinked to original sources

arXiv2025

Distribution of products of shifted primes in arithmetic progressions with increasing difference

We obtain an asymptotic formula for the number of primes $p\leq x_1$, $p\leq x_2$ such that $p_1(p_2+a)\equiv l \pmod q$ with $(a,q)=(l,q)=1$, $q\leq x^{κ_0}$, $x_1\geq x^{1-α}$, $x_2\geq x^α$, $$ κ_0=\frac{1}{2.5+θ+\varepsilon}, \quad α\in \left[(θ+\varepsilon)\frac{\ln q}{\ln x}, 1-2.5\frac{\ln q}{\ln x}\right], $$ where $θ=1/2$, if $q$ is a cube free and $θ=\frac{5}{6}$ otherwise. This is the refinement and generalization of the well-known formula of A.~A.~Karatsuba.\\ Keywords: {Dirichlet character, shifted primes, short sum of characters with primes}\\ Bibliography: 39 references

math.NT↗

arXiv2025

Sums of values of non-principal characters over shifted primes

For a nonprincipal character $χ$ modulo $D$, when $x\ge D^{\frac56+\varepsilon}$, $(l,D) = 1$, we prove a nontrivial estimate of the form $\sum_{n\le x}Λ(n)χ(n-l)\ll x\exp\left(-0.6\sqrt{\ln D}\right)$ for the sum of values of $χ$ over a sequence of shifted primes. Bibliography: 41 references.

math.NT↗

arXiv2024

Waring's problem with almost proportional summands

For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1, r}$, satisfying the conditions $$ |x_i^n-μ_iN|\le H,\qquad H\ge N^{1-θ(n,r)+\varepsilon},\qquad θ(n,r)=\frac2{(r+1)(n^2-n)}, $$ where $μ_1, \ldots, μ_r$ are positive fixed numbers, and $μ_1 + \ldots + μ_n = 1$. This result strengthens the theorem of E.M.Wright.

math.NT↗