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Large character sums: Pretentious characters and the PolyaVinogradov theorem
 J. Amer. Math. Soc
"... The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135137). For any nonprincipal Dirichlet character χ (mod q) we let M(χ): = max ..."
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The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135137). For any nonprincipal Dirichlet character χ (mod q) we let M(χ): = max
Large Character Sums
 CHARACTERS AND THE POLYAVINOGRADOV THEOREM 29
"... A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x ..."
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A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x
CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
, 2003
"... The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and GrossZagier, the task was a finite decision problem for any N. ..."
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The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and GrossZagier, the task was a finite decision problem for any N. Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work for other small class numbers, up to 100, say.” However, more than ten years later, after doing N =5, 6, 7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N ≤ 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the GoldfeldOesterlé work, which used an elliptic curve Lfunction with an order 3 zero at the central critical point, to instead consider Dirichlet Lfunctions with lowheight zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of MontgomeryWeinberger. Our method is still quite computerintensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large “exceptional modulus” of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.
Several generalizations of Weil sums
 J. Number Theory
, 1994
"... We consider several generalizations and variations of the character sum inequalities of Weil and Burgess. A number of incomplete character sum inequalities are proved while further conjectures are formulated. These inequalities are motivated by extremal graph theory with applications to problems in ..."
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We consider several generalizations and variations of the character sum inequalities of Weil and Burgess. A number of incomplete character sum inequalities are proved while further conjectures are formulated. These inequalities are motivated by extremal graph theory with applications to problems in computer science. 1 1.
Pretentiousness in analytic number theory
 J. Theor. Nombres Bordeaux
, 2009
"... Abstract. In this report, prepared specially for the program of the XXVième Journées Arithmétiques, we describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of “pretentiousness ” to help us better understand several key questions in analytic number theory ..."
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Abstract. In this report, prepared specially for the program of the XXVième Journées Arithmétiques, we describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of “pretentiousness ” to help us better understand several key questions in analytic number theory. The prime number theorem, I As a boy of 15 or 16, Gauss determined, by studying tables of primes, that the primes occur with density 1log x at around x. This translates into the guess that π(x): = #{primes ≤ x} ≈ Li(x) where Li(x):= ∫ x 2 dt log t ∼ x log x The existing data lend support to Gauss’s guesstimate: x π(x) = #{primes ≤ x} Overcount: [Li(x) − π(x)]
THE FREQUENCY AND THE STRUCTURE OF LARGE CHARACTER SUMS
"... Abstract. Let M() denote the maximum of j∑nN (n)j for a given nonprincipal Dirichlet character (mod q), and let N denote a point at which the maximum is attained. In this article we study the distribution of M()= p q as one varies over characters (mod q), where q is prime, and investigate the lo ..."
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Abstract. Let M() denote the maximum of j∑nN (n)j for a given nonprincipal Dirichlet character (mod q), and let N denote a point at which the maximum is attained. In this article we study the distribution of M()= p q as one varies over characters (mod q), where q is prime, and investigate the location of N. We show that the distribution of M()= p q converges weakly to a universal distribution , uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for 's tail. Almost all for which M() is large are odd characters that are 1pretentious. Now, M() j∑nq=2 (n)j = j2(2)j pqjL(1; )j, and one knows how often the latter expression is large, which has been how earlier lower bounds on were mostly proved. We show, though, that for most with M() large, N is bounded away from q=2, and the value of M() is little bit larger than
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"... Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms. (English summary) Methods Funct. Anal. Topology 15 (2009), no. 2, 177–187. Let (Ω, Σ, µ) be a complete finite measure space and let L 0 = L 0 (Ω) be the algebra of measurable complex valued ..."
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Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms. (English summary) Methods Funct. Anal. Topology 15 (2009), no. 2, 177–187. Let (Ω, Σ, µ) be a complete finite measure space and let L 0 = L 0 (Ω) be the algebra of measurable complex valued functions on (Ω, Σ, µ). Further, let X be a module over L 0 together with an L 0valued inner product 〈·, ·〉: X × X → L 0 such that (X, ‖ · ‖) = X is a KaplanskyHilbert module ( ‖ · ‖ is the corresponding L 0valued norm on X). L(X) denotes the algebra of all L 0linear operators on X. An algebra U ⊂ L(X) over L 0 is said to be standard if F(X) ⊂ U, where F(X) is the algebra of all L 0linear operators a on X such that aX is a finitegenerated submodule of X. Let D ⊂ X be a (bo)dense submodule (i.e., for ϕ ∈ X there exists a net (ϕα) in D such that ‖ϕ − ϕα ‖ → 0 in L 0 almost everywhere). Finally, L(D) denotes the algebra of all L 0linear operators a: D → D. This is in general an algebra of unbounded operators. Among other things the paper contains the following results on the structure of derivations and automorphisms of algebras of unbounded operators: (1) Let δ: U → L(D) be an L 0linear derivation of a standard algebra U. Then there exists an x ∈ L(D) such that δ(a) = xa − ax for all a ∈ U. (2) Let α: F(D) → F(D) be an L 0linear automorphism. Then there is an x ∈ L(D) such that x −1 ∈ L(D) and α(a) = xax −1 for all a ∈ F(D). The same result is valid for standard algebras. The results should be compared with the corresponding results valid in algebras of unbounded operators on Hilbert spaces [see K. Schmüdgen, Unbounded operator algebras and representation theory, Birkhäuser, Basel, 1990;