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Displaying similar documents to “Congruences for certain binomial sums”

On Lehmer's problem and Dedekind sums

Xiaowei Pan, Wenpeng Zhang (2011)

Czechoslovak Mathematical Journal

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Let p be an odd prime and c a fixed integer with ( c , p ) = 1 . For each integer a with 1 a p - 1 , it is clear that there exists one and only one b with 0 b p - 1 such that a b c (mod p ). Let N ( c , p ) denote the number of all solutions of the congruence equation a b c (mod p ) for 1 a , b p - 1 in which a and b ¯ are of opposite parity, where b ¯ is defined by the congruence equation b b ¯ 1 ( mod p ) . The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L -functions to study the hybrid mean value...

On a kind of generalized Lehmer problem

Rong Ma, Yulong Zhang (2012)

Czechoslovak Mathematical Journal

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For 1 c p - 1 , let E 1 , E 2 , , E m be fixed numbers of the set { 0 , 1 } , and let a 1 , a 2 , , a m ( 1 a i p , i = 1 , 2 , , m ) be of opposite parity with E 1 , E 2 , , E m respectively such that a 1 a 2 a m c ( mod p ) . Let N ( c , m , p ) = 1 2 m - 1 a 1 = 1 p - 1 a 2 = 1 p - 1 a m = 1 p - 1 a 1 a 2 a m c ( mod p ) ( 1 - ( - 1 ) a 1 + E 1 ) ( 1 - ( - 1 ) a 2 + E 2 ) ( 1 - ( - 1 ) a m + E m ) . We are interested in the mean value of the sums c = 1 p - 1 E 2 ( c , m , p ) , where E ( c , m , p ) = N ( c , m , p ) - ( ( p - 1 ) m - 1 ) / ( 2 m - 1 ) for the odd prime p and any integers m 2 . When m = 2 , c = 1 , it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.