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Character sums with subsequence sums
Periodica Mathematica Hungarica, 2007Let χ be a primitive multiplicative character modulo an integer m ≥ 1. Using some classical bounds of character sums, we estimate the average value of the character sums with subsequence sums \( T_m (\mathcal{S},\chi ) = \sum\nolimits_{\mathcal{I} \subseteq \{ 1, \ldots ,N\} } {\chi (\sum\nolimits_{i \in \mathcal{I}} {s_i } )} \) taken over all N ...
Sanka Balasuriya, Igor E. Shparlinski
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Reducing character sums to Kloosterman sums
Mathematical Notes, 2010In this paper the authors apply a bound for very short Kloosterman type sums to deduce a bound for a mean-value of short sums of Dirichlet characters. For details, define \[ S^*=\mathop{{\sum}^*}_{\chi (\bmod \;q )}\chi(n)\overline{\chi}(m) \left(\sum_{u}\alpha_u\chi(u)\right)\left(\sum_{v}\beta_v\chi(v)\right)\left|L_f(\chi)\right|^2, \] where the ...
Friedlander, J. B., Iwaniec, H.
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Canadian Journal of Mathematics, 1979
For a non-principal Dirichlet character χ modulo q,Let the Pólya-Vingradov inequality asserts that M(x) < q1/2 log q see [7]. in the opposite direction it is a trivial consequence of lemma 1 below and 1. Parseval's identity that if χ is primitive modulo q, thenWe show that on average the latter of these estimates is the more precise.THEOREM 1.
Montgomery, H. L., Vaughan, R. C.
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For a non-principal Dirichlet character χ modulo q,Let the Pólya-Vingradov inequality asserts that M(x) < q1/2 log q see [7]. in the opposite direction it is a trivial consequence of lemma 1 below and 1. Parseval's identity that if χ is primitive modulo q, thenWe show that on average the latter of these estimates is the more precise.THEOREM 1.
Montgomery, H. L., Vaughan, R. C.
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Linear Codes and Character Sums
COMBINATORICA, 2002\textit{G. Kalai} and \textit{N. Linial} [IEEE Trans. Inf. Theory 41, 1467-1472 (1995; Zbl 0831.94019)] conjectured that the size of the code with the distribution of distances near the minimal distance is exponentially small. The authors estimate the fraction of non-zero vectors of minimal weights in an \(r\cdot n\)-dimensional subspace of \(\mathbb{Z}
Linial, Nathan, Samorodnitsky, Alex
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Acta Arithmetica, 1999
Let \(\chi\) be a Dirichlet character of conductor \(p^n\) with \(p \in {\mathbb{P}}\), \(n \in {\mathbb{N}}\), and let \(f(x) = a_0 + a_1 x + \cdots + a_k x^k\) be an integral polynomial such that \(k>3\) and \((p^n,a_1,\ldots,a_k)= p^m\). Using a special iteration the author proves some general character sum estimates of type \[ p^{-(n-m)(1-1/k ...
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Let \(\chi\) be a Dirichlet character of conductor \(p^n\) with \(p \in {\mathbb{P}}\), \(n \in {\mathbb{N}}\), and let \(f(x) = a_0 + a_1 x + \cdots + a_k x^k\) be an integral polynomial such that \(k>3\) and \((p^n,a_1,\ldots,a_k)= p^m\). Using a special iteration the author proves some general character sum estimates of type \[ p^{-(n-m)(1-1/k ...
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Izvestiya: Mathematics, 2000
The paper investigates weighted character sums of type \[ \sum_{n \leq N} \tau_k(n) \chi(n+a). \] Here, \(\chi\) is a non-principal Dirichlet character modulo a prime number \(p\), \(\tau_k(n)\) the number of positive integer solutions \(x_1, \ldots , x_k\) of the equation \(x_1 \cdots x_k = n\) and \((a,p)=1\).
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The paper investigates weighted character sums of type \[ \sum_{n \leq N} \tau_k(n) \chi(n+a). \] Here, \(\chi\) is a non-principal Dirichlet character modulo a prime number \(p\), \(\tau_k(n)\) the number of positive integer solutions \(x_1, \ldots , x_k\) of the equation \(x_1 \cdots x_k = n\) and \((a,p)=1\).
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2000
In this chapter we give some basic definitions as well as some theorems (without proofs) which are needed in later chapters. Most of the notation used in the book is explained in this chapter. Notation related to p-adic numbers and functions, especially p-adic L-functions, is given in Chapter VI. The main result of this chapter is an identity given in [
Jerzy Urbanowicz, Kenneth S. Williams
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In this chapter we give some basic definitions as well as some theorems (without proofs) which are needed in later chapters. Most of the notation used in the book is explained in this chapter. Notation related to p-adic numbers and functions, especially p-adic L-functions, is given in Chapter VI. The main result of this chapter is an identity given in [
Jerzy Urbanowicz, Kenneth S. Williams
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