An elementary proof of a formula of Kuznecov for Kloosterman sums (Q752069)

The Kloosterman sums S(, ;) satisfy \[ S(n,m;c)=\sum_{d| (n,m,c)}dS(1,nm/d^ 2;c/d). \] This formula has been stated by \textit{A. Selberg} in 1938 [9. Congr. Math. Scand., 320-322 (1939; Zbl 0021.02503), see p. 37 in his Collected Papers 1 (1989; Zbl 0675.10001)]. A proof based on the sum formula of Kuznetsov is given in theorem 4 of [\textit{N. V. Kuznetsov}, Mat. Sb., Nov. Ser. 111(153), 334-383 (1980; Zbl 0427.10016)]. In the present paper the author gives a proof of this result by elementary methods.