Anatoly Karatsuba

Karatsuba found (2000) the backward relation of estimates of the values R k ( x ) {\displaystyle R_{k}(x)} with the behaviour of ζ ( s ) {\displaystyle \zeta (s)} near the line R e   s = 1 {\displaystyle Re\ s=1} . In particular, he proved that if α ( y ) {\displaystyle \alpha (y)} is an arbitrary non-increasing function satisfying the condition 1 / y ≤ α ( y ) ≤ 1 / 2 {\displaystyle 1/y\leq \alpha (y)\leq 1/2} , such that for all k ≥ 2 {\displaystyle k\geq 2} the estimate

Karatsuba obtained (2000) non-trivial estimates of sums of values of Dirichlet characters "with weights", that is, sums of components of the form χ ( n ) f ( n ) {\displaystyle \chi (n)f(n)} , where f ( n ) {\displaystyle f(n)} is a function of natural argument. Estimates of that sort are applied in solving a wide class of problems of number theory, connected with distribution of power congruence classes, also primitive roots in certain sequences.

Karatsuba developed (1993—1999) a new method of estimating short Kloosterman sums, that is, trigonometric sums of the form

Karatsuba also obtained a number of results about the distribution of zeros of ζ ( s ) {\displaystyle \zeta (s)} on «short» intervals of the critical line. He proved that an analog of the Selberg conjecture holds for «almost all» intervals ( T , T + H ] {\displaystyle (T,T+H]} , H = T ε {\displaystyle H=T^{\varepsilon }} , where ε {\displaystyle \varepsilon } is an arbitrarily small fixed positive number. Karatsuba developed (1992) a new approach to investigating zeros of the Riemann zeta-function on «supershort» intervals of the critical line, that is, on the intervals ( T , T + H ] {\displaystyle (T,T+H]} , the length H {\displaystyle H} of which grows slower than any, even arbitrarily small degree T {\displaystyle T} . In particular, he proved that for any given numbers ε {\displaystyle \varepsilon } , ε 1 {\displaystyle \varepsilon _{1}} satisfying the conditions 0 < ε , ε 1 < 1 {\displaystyle 0<\varepsilon ,\varepsilon _{1}<1} almost all intervals ( T , T + H ] {\displaystyle (T,T+H]} for H ≥ exp ⁡ { ( ln ⁡ T ) ε } {\displaystyle H\geq \exp {\{(\ln T)^{\varepsilon }\}}} contain at least H ( ln ⁡ T ) 1 − ε 1 {\displaystyle H(\ln T)^{1-\varepsilon _{1}}} zeros of the function ζ ( 1 2 + i t ) {\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}} . This estimate is quite close to the one that follows from the Riemann hypothesis.

Up to the early 1990s, the estimates of this type were known, mainly, for sums in which the number of summands was higher than m {\displaystyle {\sqrt {m}}} (H. D. Kloosterman, I. M. Vinogradov, H. Salié, L. Carlitz, S. Uchiyama, A. Weil). The only exception was the special moduli of the form m = p α {\displaystyle m=p^{\alpha }} , where p {\displaystyle p} is a fixed prime and the exponent α {\displaystyle \alpha } increases to infinity (this case was studied by A. G. Postnikov by means of the method of Vinogradov). Karatsuba's method makes it possible to estimate Kloosterman sums where the number of summands does not exceed

Karatsuba proved (1989) that the interval ( T , T + H ] {\displaystyle (T,T+H]} , H = T 27 / 82 + ε {\displaystyle H=T^{27/82+\varepsilon }} , contains at least

In 1984 Karatsuba proved, that for a fixed ε {\displaystyle \varepsilon } satisfying the condition 0 < ε < 0.001 {\displaystyle 0<\varepsilon <0.001} , a sufficiently large T {\displaystyle T} and H = T a + ε {\displaystyle H=T^{a+\varepsilon }} , a = 27 82 = 1 3 − 1 246 {\displaystyle a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}} , the interval ( T , T + H ) {\displaystyle (T,T+H)} contains at least c H ln ⁡ T {\displaystyle cH\ln T} real zeros of the Riemann zeta function ζ ( 1 2 + i t ) {\displaystyle \zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )}} .

p {\displaystyle p} -adic method of A.A.Karatsuba includes the techniques of estimating the measure of the set of points with small values of functions in terms of the values of their parameters (coefficients etc.) and, conversely, the techniques of estimating those parameters in terms of the measure of this set in the real and p {\displaystyle p} -adic metrics. This side of Karatsuba's method manifested itself especially clear in estimating trigonometric integrals, which led to the solution of the problem of Hua Luogeng. In 1979 Karatsuba, together with his students G.I. Arkhipov and V.N. Chubarikov obtained a complete solution of the Hua Luogeng problem of finding the exponent of convergency of the integral:

His textbook Foundations of Analytic Number Theory went to two editions, 1975 and 1983.

In 1971 speaking at the International conference on number theory on the occasion of the 80th birthday of Ivan Matveyevich Vinogradov, Academician Yuri Linnik noted the following:

Karatsuba developed a number of new tools, which, combined with the Vinogradov method of estimating sums with prime numbers, enabled him to obtain in 1970 an estimate of the sum of values of a non-principal character modulo a prime q {\displaystyle q} on a sequence of shifted prime numbers, namely, an estimate of the form

For most of his student and professional life he was associated with the Faculty of Mechanics and Mathematics of Moscow State University, defending a D.Sc. there entitled "The method of trigonometric sums and intermediate value theorems" in 1966. He later held a position at the Steklov Institute of Mathematics of the Academy of Sciences.

In 1966–1980, Karatsuba developed (with participation of his students G.I. Arkhipov and V.N. Chubarikov) the theory of multiple Hermann Weyl trigonometric sums, that is, the sums of the form

where P k − 1 ( u ) {\displaystyle P_{k-1}(u)} is a polynomial of degree ( k − 1 ) {\displaystyle (k-1)} , the coefficients of which depend on k {\displaystyle k} and can be found explicitly and R k ( x ) {\displaystyle R_{k}(x)} is the remainder term, all known estimates of which (up to 1960) were of the form

Karatsuba obtained a more precise estimate of R k ( x ) {\displaystyle R_{k}(x)} , in which the value α ( k ) {\displaystyle \alpha (k)} was of order k − 2 / 3 {\displaystyle k^{-2/3}} and was decreasing much slower than α ( k ) {\displaystyle \alpha (k)} in the previous estimates. Karatsuba's estimate is uniform in x {\displaystyle x} and k {\displaystyle k} ; in particular, the value k {\displaystyle k} may grow as x {\displaystyle x} grows (as some power of the logarithm of x {\displaystyle x} ). (A similar looking, but weaker result was obtained in 1960 by a German mathematician Richert, whose paper remained unknown to Soviet mathematicians at least until the mid-seventies.)

These two theorems were proved by Karatsuba in his 4th year as a basis of his 4th year project; the corresponding paper was submitted to the journal "Uspekhi Mat. Nauk" on December 17, 1958 and published in June 1960. Up to this day (2011) this result of Karatsuba that later acquired the title "the Moore-Karatsuba theorem", remains the only precise (the only precise non-linear order of the estimate) non-linear result both in the automata theory and in the similar problems of the theory of complexity of computations.

In 1957 Karatsuba proved two theorems which completely solved the Moore problem on improving the estimate of the length of experiment in his Theorem 8.

The special case H ≥ T 1 / 2 + ε {\displaystyle H\geq T^{1/2+\varepsilon }} was proven by Atle Selberg earlier in 1942. The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as T → + ∞ {\displaystyle T\to +\infty } .

Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (Russian: Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, 31 January 1937 – Moscow, Russia, 28 September 2008) was a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series.