chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2016-17-3-106-124cheb-261Research ArticleСтатьиArticleО КОЛИЧЕСТВЕ НУЛЕЙ ДЗЕТА-ФУНКЦИИ РИМАНА, ЛЕЖАЩИХ В «ПОЧТИ ВСЕХ» ОЧЕНЬ КОРОТКИХ ПРОМЕЖУТКАХ ОКРЕСТНОСТИ КРИТИЧЕСКОЙ ПРЯМОЙON NUMBER OF ZEROS OF THE RIEMANN ZETA FUNCTION THAT LIE IN «ALMOST ALL» VERY SHORT INTERVALS OF NEIGHBORHOOD OF THE CRITICAL LINEТамДо ДыкTamDo Duc

аспирант

postgraduate

doductam140189@gmail.comБелгородский государственный национальный исследовательский университетРоссияBelgorod National Research UniversityRussian Federation201612122016173106124Copyright © Там Д., 20162016Там Д.Tam D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/261

Центральной проблемой аналитической теории чисел является доказательство (или опровержение) гипотезы Римана. К настоящему времени она не решена.

В 1985 году А. А. Карацуба доказал, что при любом \(0<\varepsilon<0,001\), \(0,5<\sigma\leq 1\), \(T>T_0(\varepsilon)>0\) и \(H=T^{27/82+\varepsilon}\) в прямоугольнике с вершинами \(\sigma+iT\), \(\sigma+i(T+H)\), \(1+i(T+H)\), \(1+iT\) содержится не больше, чем \(cH/(\sigma-0,5)\) нулей функции \(\zeta(s)\). Тем самым А.А. Карацуба существенно усилил классическую теорему Дж. Литтлвуда. Для индивидуального прямоугольника существенно уменьшить величину \(H\) не удается. Однако решая эту задачу <<в среднем>>, Л.В. Киселева в 1989 году доказала, что для <<почти всех>> \(T\) из промежутка \([X,X+X^{11/12+\varepsilon}]\), \(X>X_0(\varepsilon)\), для которых в прямоугольнике с вершинами \(\sigma+iT\), \(\sigma+i(T+X^\varepsilon)\), \(1+i(T+X^\varepsilon)\), \(1+iT\) содержится не больше, чем \(O(X^\varepsilon/(\sigma-0,5))\) нулей функции \(\zeta(s)\).

В нашей статье получен результат подобного рода, но только для <<почти всех>> \(T\) из промежутка \([X,X+X^{7/8+\varepsilon}]\).

Proof (or disproof) of the Riemann hypothesis is the central problem of analytic number theory. By now it has not been solved.

In 1985 Karatsuba proved that for any \( 0 <\varepsilon <0,001 \), \( 0,5 <\sigma \leq 1 \), \( T> T_0 (\varepsilon)> 0 \) and \( H = T ^ { 27/82 + \varepsilon} \) in the rectangle with vertices \( \sigma + iT \), \( \sigma + i (T + H) \), \( 1 + i (T + H) \), \( 1 + iT \) contains no more than \( cH / (\sigma-0,5) \) zeros of \( \zeta (s) \). Thereby A.A. Karatsuba significantly strengthened the classical theorem J. Littlewood's.

Decrease in magnitude of \(H\) for individual rectangle has not been obtained. However, by solving this problem <<on average>>, in 1989 L.V. Kiseleva proved that for <<almost all>> \( T \) in the interval \( [X, X + X ^ {11/12 + \varepsilon}] \), \( X> X_0 (\varepsilon) \) in rectangle with vertices \( \sigma + iT \), \( \sigma + i (T + X ^ \varepsilon) \), \( 1 + i (T + X ^ \varepsilon) \), \( 1 + iT \) contains no more than \( O (X ^ \varepsilon / (\sigma-0,5)) \) zeros of \( \zeta (s) \).

In this article, we obtain a result of this kind, but for <<almost all >> \( T \) in the interval \( [X, X + X ^ {7/8 + \varepsilon}] \).

дзета-функциянетривиальные нуликритическая прямаяzeta functionnon-trivial zeroscritical lineReferencesРиман Б. Сочинения. М.–Л.: ОГИЗ, 1948. 479 c.Riemann, B. 1948, “Sochineniya.” (Russian) [The works], OGIZ, Moskva–Leningrad. 479 p.Hardy G. Н. Sur les zeros de la fonctionHardy, G. Н. 1914, “Sur les zeros de la fonctionHardy, G. H. & Littlewood, J. E. 1921. “The zeros of Riemann’s zeta-function on the critical line”, Mathematische Zeitschrift, vol. 10, pp. 283–317.Hardy, G. H. & Littlewood, J. E. 1921. “The zeros of Riemann’s zeta-function on the critical line”, Mathematische Zeitschrift, vol. 10, pp. 283–317.Littlewood, J. E. 1924, “On the zeros of the Riemann zeta-function,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 22. pp 295-318 doi:10.1017/S0305004100014225Littlewood, J. E. 1924, “On the zeros of the Riemann zeta-function,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 22. pp 295-318 doi:10.1017/S0305004100014225Selberg, A. 1942, “On the zeros of Riemann’s zeta-function”, Skr. Norske. Vid. Akad Oslo, vol. 10, pp. 1–59.Selberg, A. 1942, “On the zeros of Riemann’s zeta-function”, Skr. Norske. Vid. Akad Oslo, vol. 10, pp. 1–59.Levinson, N. 1974, “More than one third of the zeros of Riemann’s zeta-function are onLevinson, N. 1974, “More than one third of the zeros of Riemann’s zeta-function are onKaratsuba, A. A. 1981, “On the distance between consecutive zeros of the Riemann zeta function that lie on the critical line”, Trudy Mat. Inst. Steklov, vol. 157, pp. 49–63. (Russian)Karatsuba, A. A. 1981, “On the distance between consecutive zeros of the Riemann zeta function that lie on the critical line”, Trudy Mat. Inst. Steklov, vol. 157, pp. 49–63. (Russian)Karatsuba, A. A. 1984, “On the zeros of the functionKaratsuba, A. A. 1984, “On the zeros of the functionKaratsuba, A. A. 1984, “The distribution of zeros of the functionKaratsuba, A. A. 1984, “The distribution of zeros of the functionKaratsuba, A. A. 1985, “Zeros of the Riemann zeta function on the critical line”, Trudy Mat. Inst. Steklov, vol. 167 , pp. 167–178. (Russian)Karatsuba, A. A. 1985, “Zeros of the Riemann zeta function on the critical line”, Trudy Mat. Inst. Steklov, vol. 167 , pp. 167–178. (Russian)Karatsuba, A. A. 1985, “On the real zeros of the functionKaratsuba, A. A. 1985, “On the real zeros of the functionKaratsuba, A. A. 1985, “The Riemann zeta function and its zeros”, Uspekhi Mat. Nauk, vol. 40, no 5, pp. 23–82. (Russian)Karatsuba, A. A. 1985, “The Riemann zeta function and its zeros”, Uspekhi Mat. Nauk, vol. 40, no 5, pp. 23–82. (Russian)Karatsuba, A. A. 1986, “On the zeros of the functionKaratsuba, A. A. 1986, “On the zeros of the functionKaratsuba, A. A. 1992, “On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line”, Izv. Akad. Nauk SSSR. Ser. Math., vol 56, no 2, pp. 372–397. (Russian)Karatsuba, A. A. 1992, “On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line”, Izv. Akad. Nauk SSSR. Ser. Math., vol 56, no 2, pp. 372–397. (Russian)Karatsuba, A. A. 1992, “A refinement of theorems on the number of zeros lying on intervals of the critical line of certain Dirichlet series”, Uspekhi Mat. Nauk, vol. 47, no 2, pp. 193–194. (Russian)Karatsuba, A. A. 1992, “A refinement of theorems on the number of zeros lying on intervals of the critical line of certain Dirichlet series”, Uspekhi Mat. Nauk, vol. 47, no 2, pp. 193–194. (Russian)Karatsuba, A. A. 1992 “On the zeros of a special type of function connected with Dirichlet series”, Math. USSR-Izv., vol. 38, no 3, pp. 471–502.Karatsuba, A. A. 1992 “On the zeros of a special type of function connected with Dirichlet series”, Math. USSR-Izv., vol. 38, no 3, pp. 471–502.Kiseleva, L. V. 1988, “The number of zeros of the functionKiseleva, L. V. 1988, “The number of zeros of the functionKiseleva, L.V. 1989 “On the zeros of the functionKiseleva, L.V. 1989 “On the zeros of the functionTam, D. D. 2015 “On the zeros of the Rieman zeta function, lying in almost all short intervals of the critical line.” Chebyishovski Sbornhik. vol. 17, no 1, pp. 71–89.Tam, D. D. 2015 “On the zeros of the Rieman zeta function, lying in almost all short intervals of the critical line.” Chebyishovski Sbornhik. vol. 17, no 1, pp. 71–89.Voronin, S. V. & Karatsuba, A. A. 1994, “Zeta-funkcia Rimana.” (Russian) [The Riemann zeta-function], Fizmatlit, Moscow, 376 p.Voronin, S. V. & Karatsuba, A. A. 1994, “Zeta-funkcia Rimana.” (Russian) [The Riemann zeta-function], Fizmatlit, Moscow, 376 p.Titchmarsh, E. K. 1953, “Teoriya dzeta-funkcii Rimana.” (Russian)[Теория дзета-функции Римана], Mir, Moscow, 409 p.Titchmarsh, E. K. 1953, “Teoriya dzeta-funkcii Rimana.” (Russian)[Теория дзета-функции Римана], Mir, Moscow, 409 p.Malysev, A. V. 1962, “On the representation of integers by positive quadratic forms.” (Russian) Trudy Mat. Inst. Steklov, vol. 65, pp. 3-212.Malysev, A. V. 1962, “On the representation of integers by positive quadratic forms.” (Russian) Trudy Mat. Inst. Steklov, vol. 65, pp. 3-212.Karatsuba, A. A. 1983, “Osnovui analiticheskoi teorii chisel.” (Russian) [Fundamentals of analytic number theory], Nauka, Мoscow, 240 p.Karatsuba, A. A. 1983, “Osnovui analiticheskoi teorii chisel.” (Russian) [Fundamentals of analytic number theory], Nauka, Мoscow, 240 p.

The authors declare that there are no conflicts of interest present.