arXiv subjects
Publications and source records attributed to M. A. Korolev.
We give an elementary proof of some identities that express the squares of Riemann zeta function at integer points in terms of the series involving hyperbolic functions, digamma function, Bernoulli numbers etc. In this version, inaccuracies in the text have been corrected and one of the bibliographic references has been updated.
If $d$ is not a perfect square, we define $T(d)$ as the length of the minimal period of the simple continued fraction expansion for $\sqrt{d}$. Otherwise, we put $T(d) = 0$. In the recent paper (2024), F.Battistoni, L.Grenié and G.Molteni established (in particular) an upper bound for the second moment of $T(d)$ over the segment $x α\sqrt{x}$. In this paper, we improve slightly this result of three authors. In this version, we improve the estimate of the remainder term in the main asymptotic formula, correct some misprints and add an important remark made by F.Battistoni, L.Grenié and G.Molteni. This remark concerns the explicit value of the constant in the main term of asymptotics.
In the paper, we establish a new estimate for Kloosterman sum over primes with respect to an arbitrary modulus $q$. This estimate together with some recent results of the second author are applied to the problem of solvability of the congruence \[ g(p_{1})\, + \,\ldots\, + \,g(p_{k}) \,\equiv\, m\pmod{q} \] in prime variables $p_{1},\ldots, p_{k}\le N$, $N\le q^{1-δ}$. Here $g(x)\,\equiv\,a\overline{x}+bx\pmod{q}$, where $a,b,m$ and $k\ge 3$ are arbitrary integers, $(ab,q)=1$. The main result of the paper gives an asymptotic formula for the number of solutions for the case when $q$ is coprime to $6$. In this version, we correct some typos and small errors.
We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2πi}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad p\overline{p}\equiv 1\pmod{q}, \] which is non-trivial in the case when $q^{\,3/4+\varepsilon}\leqslant X\ll q^{\,3/2}$. We also apply this estimate to the proof of solvability of some congruences with inverse prime residues $\pmod{q}$.
We obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius function and also by the generalised divisor function. These trace functions include Kloosterman sums and several other classical number theoretic objects. Our bounds are nontrivial for intervals of length at least $p^{1/2+\varepsilon}$ with an arbitrary fixed $\varepsilon >0$, which is shorter than the length at least $p^{3/4+\varepsilon}$ in the case of the Mobius function and at least $p^{2/3+\varepsilon}$ in the case of the divisor function required in recent results of {É}.~Fouvry, E.~Kowalski and P.~Michel (2014) and E.~Kowalski, P. ~Michel and W.~Sawin (2018), respectively.
Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, χ$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{\frac12+\varepsilon}\ll N\ll q^A$. In this paper, we shall prove that $$ \sum_{n\leq N}f(n)χ(n+a)\ll N\frac{\log\log q}{\log q} $$ and that $$ \sum_{n\leq N}f(n)χ(n+a_1)\cdotsχ(n+a_t)\ll N\frac{\log\log q}{\log q}, $$ where $t\geq 2, a_1, \ldots, a_t$ are distinct integers modulo $q$.
The series of some new estimates for the sums of the type \[ S_{q}(x;f)\,=\,\mathop{{\sum}'}\limits_{n\leqslant x}f(n)e_{q}(an^{*}+bn) \] is obtained. Here $q$ is a sufficiently large integer, $\sqrt{q}(\log{q})\!\ll\!x\leqslant q$, $a,b$ are integers, $(a,q)=1$, $e_{q}(v) = e^{2πiv/q}$, $f(n)$ is a multiplicative function, $nn^{*}\equiv 1 \pmod{q}$ and the prime sign means that $(n,q)=1$.
We prove some new bounds for the maximum of Riemann zeta-function on very short segments of the critical line. All the theorems are based on the Riemann hypothesis.
In this paper, we obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, the conditional solution of one problem of A.A.Karatsuba is given. Some typos of the previous versions are corrected (in particular, the important remark of Prof. Yan Fyodorov is taken into account). The reference to the grant of Russian Scientific Fund is also added.
In comparison with the previous version of this paper, the Introduction is slightly changed and some minor typos are deleted. All results are unchanged.
Some statements concerning the distribution of imaginary parts of zeros of the Riemann zeta\,-function are established. These assertions are connected with so\,-called `Gram's law' or `Gram's rule'. In particular, we give a proof of several Selberg's formulae stated him without proof in his paper `The Zeta Function and the Riemann Hypothesis' (1946), and some of their equivalents.
We study an asymptotic behavior of the sum $\sum\limits_{n\le x}\frac{\D τ(n)}{\D τ(n+a)}$. Here $τ(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.
From the data analysis we defined distribution function against the population on the level of various structure units, namely regions, federal districts and the country on the whole. We have studied peculiarities of the distribution function deformation due to the structure units' enlargement. Using the master equation in the continuous approximation, we obtain the Fokker-Plank equation for the distribution function with symmetric transition rates. In addition, we offer a model where transitions only between neighbour states are possible. Moreover in this case it is proposed the in and out transition probability rate for any states are different We define condition for the formal equivalence of both models to the problem that is described by the stochastic differential equation with additive and multiplicative white noises. We have analyzed the corresponding Fokker-Plank equation in the Ito and Stratonovich calculus, and obtained a solution of the Tsallis distribution type. We demonstrate a comparison of theoretical results with those of the data analysis on the level a cumulative distributions. We have used the maximum likelihood method for fitting. The obtained results allow us to attribute the specific value of the Tsallis distribution nonextensivity parameter to the empiric curves.