M. A. Korolev

• Russian Academy of Sciences

В работе исследуется распределение суммарной длины хорд единичной окружности, соединяющих вершины правильного $q$-угольника с номерами $k$ и $ak^{2} (\operatorname{mod} q)$, $k=1,2,…, q$, в случае, когда $q$ - простое число, а величина $a$ пробегает полную систему вычетов по модулю $q$. Библиография: 13 названий.

By the Ostrowski theorem, the Riemann zeta-function \(\zeta (s)\) does not satisfy any algebraic-differential equation. Voronin proved that the function \(\zeta (s)\) does not satisfy algebraic-differential equation with continuous coefficients. In the paper, a joint generalization of the Voronin theorem is given, i. e., that a collection \((\zeta...

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z(s+iτ), τ∈R, of the modified Mellin transform Z(s) of the square of the Riemann zeta-function ζ(1/2+it). More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z(s+iτ), which approximate a given analy...

Доказано, что максимум модуля дзета-функции Римана $\zeta (s)$ при изменении $s = 0.5+it$ на очень коротких отрезках критической прямой неограниченно возрастает, причем для скорости роста получена явная нижняя оценка. Этот основной результат работы является улучшением результата второго автора (2014), согласно которому данный максимум с ростом $t$...

In the paper, we consider the simultaneous approximation of a collection of analytic functions by a collection of shifts of the Riemann zeta-function $(\zeta(s+it_\tau^{\alpha_1}), \dots, \zeta(s+it_\tau^{\alpha_r}))$, where $t_\tau$ is the Gram function and $\alpha_1, \dots, \alpha_r$ are different positive numbers. It is obtained that the set of...

В работе исследуется "корреляционная" функция $\mathcal{K}_{P} = \mathcal{K}_{P}(T;H,U)$ остаточного члена $P(t)$ в проблеме круга, т. е. интеграл от произведения $P(t)P(t+U)$ по промежутку $(T,T+H]$, $1\le U, H\le T$. Случай малых значений $U$, $1\le U\ll \sqrt{T}$, был фактически изучен М. Ютилой в 1984 г.; при этом оказывается, что для всех указ...

We refine a bound for a short Kloosterman sum with a prime modulus using the so-called Vinogradov sieve. The number of terms in the sum can be less than an arbitrarily small fixed power of . Bibliography: 26 titles.

За счет применения так называемого решета И. М. Виноградова уточняется оценка короткой суммы Клоостермана по простому модулю $q$. Число слагаемых в такой сумме может быть меньшим сколь угодно малой фиксированной степени $q$. Библиография: 26 названий.

В настоящей заметке предложен новый способ вывода теоретико-числовых тождеств, который применяется к доказательству многомерного аналога одного из тождеств С. Рамануджана. Предлагаемый способ позволяет получать новые представления в виде бесконечных рядов для числа $\pi$, значений дзета-функции Римана и $L$-рядов Дирихле в целых точках. Библиографи...

Исследуется задача о разрешимости сравнения $g(p_1)+…+g(p_k)\equiv m\pmod {q}$ в простых числах $p_1,…,p_k\leq N$, $N\leq q^{1-\gamma }$, $\gamma >0$. Здесь $g(x)\equiv a\overline {x}+bx\pmod {q}$, $\overline {x}$ - обратный к $x$ вычет, т.е. $\overline {x}x\equiv 1\pmod {q}$, $q\geq 3$, $a$, $b$, $m$ и $k\geq 3$ - произвольные целые числа, причем...

In the paper, an analogue of the Gram points used in the theory of the Riemann zeta-function is introduced for zeta-functions of normalized Hecke-eigen cusp forms of weight κ. Some analytic properties of those points are studied, and the first ten Gram points for κ⩽12 are calculated. The main attention is devoted to the universality of zeta-functio...

The paper is a continuation of Korolev and Laurinčikas (Aequ Math 93:859–873, 2019), where theorems on the approximation of analytic functions by shifts \(\zeta (s+iht_k)\), \(h>0\), \(k\in {\mathbb {N}}\), and \(t_k\) are the Gram points, were obtained. In this paper, it is proved, that the set of shifts \(\zeta (s+iht_k)\) has a positive density...

В работе получена новая оценка суммы Клоостермана с простыми числами по произвольному модулю $q$, длина $X$ которой удовлетворяет условиям $$ q^{1/2+\varepsilon}\le X\ll q^{3/2}. $$ Эта оценка уточняет результаты, полученные ранее Э. Фуври, И. Е. Шпарлинским (2011) и первым автором (2018, 2019). Библиография: 10 названий.

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 $$\begin{aligned} \sum \limits _{p\leqslant X,\;p\,\not \mid q}\exp {\biggl (\frac{2\pi i}{q}\bigl (a\overline{p}+bp\bigr )\,\biggr )},\quad (ab,q)=1,\quad p\overline{p}\equiv 1\pmod {q}, \end{aligned}$$which...

We give a new estimate of the error term in the asymptotic formula for the second moment of first derivative of Hardy’s function Z(t). This estimate improves the previous result of R.R. Hall.

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}\l...

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 \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi i}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad p\overline{p}\equiv 1\pmod{q}, \] which is non-trivial in the case when $q^{\,...

In the talk, we present several new theorems concerning the sums of values of Riemann zeta-function and Hardy's Z-function over Gram points.

In the talk, we present several new theorems concerning the sums of values of Riemann zeta-function and Hardy's Z-function over Gram points

The Gram points \(t_n\) are defined as solutions of the equation \(\theta (t)=(n-1)\pi \), \(n\in \mathbb {N}\), where \(\theta (t)\), \(t>0\), denotes the increment of the argument of the function \(\pi ^{-s/2}\Gamma \left( \frac{s}{2}\right) \) along the segment connecting the points \(s=\frac{1}{2}\) and \(s=\frac{1}{2}+it\). In the paper, theor...

We obtain an expression for the density of the distribution of the lengths of arcs connecting neighbouring rational points on the unit circle with denominators not exceeding a given bound.

В работе получено выражение для плотности распределения длин дуг, соединяющих соседние рациональные точки единичной окружности со знаменателями, не превосходящими заданной границы. Библиография: 10 наименований.

В работе получена новая оценка суммы Клоостермана с простыми числами по простому модулю $q$, число слагаемых в которой может иметь порядок $q^{0.5+\varepsilon}$. Эта оценка уточняет результаты, полученные ранее Ж. Бургейном (2005) и Р. Бейкером (2012). Библиография: 22 названия.

Let S(t):=[Formula presented]arg⁡ζ([Formula presented]+it). We prove that, for T27/82+ε⩽H⩽T, we have mes{t∈[T,T+H]:S(t)>0}=[Formula presented]+O([Formula presented]), where the O-constant is absolute. A similar formula holds for the measure of the set with S(t)<0, where logk⁡T=log⁡(logk−1⁡T). This result is derived from an asymptotic formula for th...

A new estimate of the Kloosterman sum with primes modulo a prime number q is obtained, in which the number of summands can be of order q0.5+ε. This estimate refines results obtained earlier by J. Bourgain (2005) and R. Baker (2012).

We obtain an asymptotic formula for the average number of divisors of the quadratic form A(x, y, z) = xy + xz + yz, where x, y, and z run through prime numbers from the interval X <x,y,z ≤ 2X.

Let $S(t) \;:=\; \frac{\displaystyle 1}{\displaystyle \pi}\arg \zeta(\frac{1}{2} + it)$. We prove that, for $T^{\,27/82+\varepsilon} \le H \le T$, we have $$ {\rm mes}\Bigl\{t\in [T, T+H]\;:\; S(t)>0\Bigr\} = \frac{H}{2} + O\left(\frac{H\log_3T}{\varepsilon\sqrt{\log_2T}}\right), $$ where the $O$-constant is absolute. A similar formula holds for th...

В работе получен ряд новых оценок для сумм вида $$ S_{q}(x;f)=\mathop{{\sum}'}_{n\le x}f(n)e_{q}(an^*+bn), $$ в которых $q$ - достаточно большое целое число, $\sqrt{q} (\log{q})\ll x\le q$, $a$, $b$ - целые, причем $(a,q)=1$, $e_{q}(v) = e^{2\pi iv/q}$, $f(n)$ - мультипликативная функция, удовлетворяющая некоторым ограничениям, $nn^*\equiv 1 \pmod{...

In the paper, the explicit form of distribution function for the lengths of arcs connecting neighbouring rational points on the unit circle whose denominators do not exceed given value, is given.

В работе получено новое элементарное доказательство оценки неполной суммы Клоостермана по простому модулю $q$. Наряду с полученной Ж. Бургейном (2005) оценкой двойной суммы Клоостермана специального вида, оно приводит к элементарному выводу оценки суммы Клоостермана с простыми числами в случае, когда длина суммы имеет порядок $q^{0.5+\varepsilon}$,...

Analogs are obtained of the asymptotic Riemann-Siegel formulas for the first and second order derivatives of the Hardy function Z(t) and the Riemann zeta function on the critical line.

For an arbitrary composite modulus q a bound is obtained for a short Kloosterman sum with primes whose length exceeds q⁷/10+ε. This bound improves the previous result by Fouvry and Shparlinski, which holds for sums of length at least q³/4+ε . © 2018 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

A new elementary proof of an estimate for incomplete Kloosterman sums modulo a prime q is obtained. Along with Bourgain’s 2005 estimate of the double Kloosterman sum of a special form, it leads to an elementary derivation of an estimate for Kloosterman sums with primes for the case in which the length of the sum is of order q0.5+ε, where ε is an ar...

Получена оценка короткой суммы Клоостермана по произвольному составному модулю $q$ с простыми числами, длина которой превышает $q^{7/10+\varepsilon}$. Эта оценка уточняет предыдущий результат Э. Фуври и И. Е. Шпарлинского, который справедлив для сумм с длиной, не меньшей $q^{3/4+\varepsilon}$. Библиография: 23 названия.

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 l...

An asymptotic formula, with a new estimate for the remainder term, is obtained for the number of solutions of a symmetric Diophantine equation involving reciprocal integers. Bibliography: 7 titles. © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd..

В работе получена асимптотическая формула для числа решений симметричного диофантова уравнения, содержащего обратные величины, с новой оценкой остаточного члена. Библиография: 7 названий.

A sharpened lower bound is obtained for the number of solutions to an inequality of the form α ≤ {(an̅ + bn)/q} < β, 1 ≤ n ≤ N, where q is a sufficiently large prime number, a and b are integers with (ab, q) = 1, nn̅ ≡ 1 (mod q), and 0 ≤ α < β ≤ 1. The length N of the range of the variable n is of order qε, where ε > 0 is an arbitrarily small fixed...

An overview is given of the scientific results obtained by Anatolii Alekseevich Karatsuba between the early 1990s and 2008.

The work is devoted to generalized Kloosterman sums modulo a prime, i.e., trigonometric sums of the form \(\sum\nolimits_{p \leqslant x} {\exp \left\{ {2\pi i\left( {a\bar p + {F_k}\left( p \right)} \right)/q} \right\}} \) and \(\sum\nolimits_{n \leqslant x} {\mu \left( n \right)\exp \left\{ {2\pi i\left( {a\bar n + {F_k}\left( n \right)} \right)/q...

В статье с помощью метода А. А. Карацубы получены новые оценки сумм Клоостермана по простому модулю, которые при некоторых ограничениях на число слагаемых являются более точными, чем аналогичные оценки, найденные ранее. Библиография: 13 названий.

We obtain new estimates for the maximum and minimum of the argument of the Riemann zetafunction on very short segments of the critical line. These results are based on the Riemann hypothesis.

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ 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...

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ 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...

Using the Karatsuba method, we obtain new estimates for Kloosterman sums modulo a prime, which, under certain constraints on the number of summands, are sharper than similar estimates found earlier.

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, $(a,q)=1$, $e_{q}(v) = e^{2\pi iv/q}$, $f(n)$ is a multiplicative function, $nn^{*}\equiv 1 \pmod...

Using Karatsuba's method, we obtain estimates for Kloosterman sums modulo a prime, in which the number of terms is less than an arbitrarily small fixed power of the modulus. These bounds refine similar results obtained earlier by Bourgain and Garaev. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

An asymptotic formula is obtained for the number of solutions to a symmetric Diophantine equation with reciprocals, and its applications to problems related to the distribution of values of short Kloosterman sums are presented.

В работе с помощью метода Карацубы получены оценки сумм Клоостермана по простому модулю, число слагаемых в которых меньше любой сколь угодно малой фиксированной степени модуля. Эти оценки уточняют аналогичные результаты, полученные ранее Ж. Бургейном и М. З. Гараевым. Библиография: 16 наименований.

We obtain the estimate of incomplete Kloosterman sum to powerful modulus $q$. The length $N$ of the sum lies in the interval $e^{c(\log{q})^{2/3}}\le N\le \sqrt{q}$.

Square-root bounds for the quadratic Gaussian sums are given with explicit constants.

In January, 2014, the I’st one-dayinternational “Conference to the Memory of A.A. Karatsuba on Number Theory and Applications” took place in Steklov Mathematical Institute of Russian Academy of sciences. The aims of this conferencewere presentationof newandimportantresultsin differentbranches of number theory (especially in branches connected with w...

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.

In this paper, we prove the existence of a large set of Gram points such that the values are 'anomalously' close to zero. A lower bound for the negative 'discrete' moment of the Riemann zeta-function on the critical line is also given.

We prove a number of new results related to Gram's law in the theory of the Riemann zeta-function that reflect irregularity in the distribution of the ordinates of the complex zeros of that function. Results are obtained on the distribution of pairs, triples, quadruples,⋯ of adjacent ordinates of such zeros that simultaneously do not obey Gram's la...

In the paper we obtain the asymtotic number of integral quadratic polynomials with bounded heights and discriminants as the upper bound of heights tends to infinity.

We prove under RH the existence of a very large positive and negative values of the argument of the Riemann zeta function on a very short intervals.

In the paper, we obtain a new proof of Selberg formulae related to Gram's law in the theory of the Riemann zeta function. Bibliography: 12 titles.

“The paper contains the formulations of some new results related to Gram’s law in the theory of the Riemann zeta-function and describing the irregularity in the distribution of complex zeros of this function. Namely, we obtain some results related to the distribution of pairs, triples, quadruples etc. of the neighbouring ordinates of such zeros tha...

We prove an assertion of Selberg concerning Gram's rule and the distribution of zeros of the Riemann zeta function. We also prove some equivalent assertions.

A number of new results related to Gram’s law in the theory of the Riemann zetafunction are proved. In particular, a lower bound is obtained for the number of ordinates of the zeros of the zeta-function that lie in a given interval and satisfy Gram’s law.

We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.

Errata to the paper by M A Korolev 'On large distances between consecutive zeros of the Riemann zeta-function' Izv. Ross. Akad. Nauk Ser. Mat. 72:2, 2008, 91-104. English transl.: Izv. Math., 72, 2008, 291-304.

In comparison with the previous version of this paper, the Introduction is slightly changed and some minor typos are deleted. All results are unchanged.

We study the behaviour of the quantities and , that is, the number of th power residues in the reduced and complete residue systems modulo a composite number , respectively, where is an arbitrary fixed number. In particular, we prove asymptotic formulae for the sum functions and of these quantities.

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...

We study an asymptotic behavior of the sum $\sum\limits_{n\le x}\frac{\D \tau(n)}{\D \tau(n+a)}$. Here $\tau(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.

We obtain a new estimate for Kloosterman sums with weights in which the number of summands is significantly less than any arbitrarily small fixed power of the modulus.

We prove a number of new assertions related to the zeros of the Riemann zeta function \zeta(s) and to the so-called Gram law.