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On Lehmer's problem and Dedekind sums

Xiaowei Pan, Wenpeng Zhang (2011)

Czechoslovak Mathematical Journal

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 problem involving...

On the 2 k -th power mean of L ' L ( 1 , χ ) with the weight of Gauss sums

Dongmei Ren, Yuan Yi (2009)

Czechoslovak Mathematical Journal

The main purpose of this paper is to study the hybrid mean value of L ' L ( 1 , χ ) and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value χ χ 0 | τ ( χ ) | | L ' L ( 1 , χ ) | 2 k of L ' L and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.

On the Brun-Titchmarsh theorem

James Maynard (2013)

Acta Arithmetica

The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size x / ( q 1 / 2 ϕ ( q ) ) when logx/logq ≥ 8 and is bounded....

On the mean value of the generalized Dirichlet L -functions

Rong Ma, Yuan Yi, Yulong Zhang (2010)

Czechoslovak Mathematical Journal

Let q 3 be an integer, let χ denote a Dirichlet character modulo q . For any real number a 0 we define the generalized Dirichlet L -functions L ( s , χ , a ) = n = 1 χ ( n ) ( n + a ) s , where s = σ + i t with σ > 1 and t both real. They can be extended to all s by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet L -functions especially for s = 1 and s = 1 2 + i t , and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.

On the size of L(1,χ) and S. Chowla's hypothesis implying that L(1,χ) > 0 for s > 0 and for real characters χ

S. Louboutin (2013)

Colloquium Mathematicae

We give explicit constants κ such that if χ is a real non-principal Dirichlet character for which L(1,χ) ≤ κ, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that L(s,χ) > 0 for s > 0. These constants are larger than the previous ones κ = 1- log 2 = 0.306... and κ = 0.367... we obtained elsewhere.

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