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For relatively prime positive integers u 0 and r, we consider the arithmetic progression {u k := u 0 + kr} n k=0. Define Ln := lcm{u 0 , u 1 ,. .. , un} and let a ≥ 2 be any integer. In this paper, we show that, for integers α, r ≥ a and n ≥ 2αr, we have Ln ≥ u 0 r α+a−2 (r + 1) n. In particular, letting a = 2 yields an improvement to the best previous(More)
Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetical function g k,a,b for any positive integer n by g k,a,b (n) := (b+na)(b+(n+1)a)···(b+(n+k)a) lcm(b+na,b+(n+1)a,··· ,b+(n+k)a). Letting a = 1 and b = 0, then g k,a,b becomes the arithmetical function introduced previously by Farhi. Farhi proved g k,1,0 is periodical and k! is a period. divides(More)
Let F q be the finite field of q elements with characteristic p and F q m its extension of degree m. Fix a nontrivial additive character ψ of F p. For any Laurent polynomial −1 n ], we form the exponential sum S * m (f) := The corresponding L-function L * (f, t) is defined by L * (f, t) := exp (∞ ∑ m=0 S * m (f) t m m). The corresponding L-function L(f, t)(More)
Let p be a prime. We obtain good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli numberˆB n n when n is divisible by p − 1. As an application, we give a simple proof of Clarke's 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo p for the divided universal Bernoulli numbers for the(More)