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А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
May 26, 2017 12:45–13:15, Moscow, Steklov Mathematical Institute  On the irreducible solutions of the equation with inverses

S. V. Konyagin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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Abstract: Consider the following symmetric Diophantine equation
$$\frac{1}{x_{1}}+\ldots + \frac{1}{x_{r}} = \frac{1}{x_{r+1}}+\ldots + \frac{1}{x_{2r}},\qquad (1)$$
where $r\geqslant 3$, and the variables $x_{1},\ldots, x_{2r}$ run through the segment $[1,N]$. Such equations appear in te problems connected with the estimates of incomplete Kloosterman sums.
The solution of (1) is called irreducible if any component from the set $x_{1},\ldots, x_{r}$ is not contained in the set $x_{r+1},\ldots, x_{2r}$. The following assertions holds true.
Theorem 1. Let $N,r\geqslant 3$. Then the number $J_{r}(N)$ of irreducible solutions of the equation (1) with positive integer variables $1\leqslant x_{1},\ldots, x_{2r}\leqslant N$ obeys the estimate:
$$J_{r}(N)<e^{(3r)^{3}-90}N^{ r - r/(2(2r-1))} (\frac{\ln{N}}{r}+9)^{10r^{2}}\exp{(\frac{26r^{3/2}\sqrt{\ln{N}}}{\ln{(r\ln{N})}})}.$$

The estimate of Theorem 1 allows one to derive an asymptotic formula for the whole number $I_{r}(N)$ of solutions of the equation (1) with integer variables $1\leqslant x_{1},\ldots, x_{2r}\leqslant N$. Namely, we have
Theorem 2. Let $N,r\geqslant 3$. Then the number $I_{r}(N)$ satisfies the relation
$$I_{r}(N) = r!N^{r}(1 + \delta_{r}(N)),$$
where
$$|\delta_{r}(N)|\leqslant e^{(3r)^{3}-90}N^{- r/(2(2r-1))}(\frac{\ln{N}}{r}+9)^{10r^{2}}\exp{(\frac{26r^{3/2}\sqrt{\ln{N}}}{\ln{(r\ln{N})}})}.$$
In the talk, we briefly describe main ideas that allow one to derive the above theorems and some other assertions concerning the number of solutions of the equation (1).

Language: English

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