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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
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
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.
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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;
THE SMOOTHED PÓLYA–VINOGRADOV INEQUALITY
"... Abstract. Let χ be a primitive Dirichlet character to the modulus q. Let Sχ(M, N) = ∑ M<n≤N χ(n). The PólyaVinogradov inequality states that Sχ(M, N)  ≪ √ q log q. The smoothed Pólya–Vinogradov inequality, recently introduced by Levin, Pomerance and Soundararajan, is a numerically useful ve ..."
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Abstract. Let χ be a primitive Dirichlet character to the modulus q. Let Sχ(M, N) = ∑ M<n≤N χ(n). The PólyaVinogradov inequality states that Sχ(M, N)  ≪ √ q log q. The smoothed Pólya–Vinogradov inequality, recently introduced by Levin, Pomerance and Soundararajan, is a numerically useful version of the Pólya–Vinogradov inequality that saves a log q factor. The smoothed Pólya–Vinogradov inequality has been used to settle a conjecture of Brizolis, namely that for every prime p> 3, there is a primitive root g and an integer x ∈ [1, p − 1] such that gx ≡ x mod p. It has also been used to improve the best known numerically explicit upper bound on the least inert prime in a real quadratic field. In this paper we will prove a smoothed Pólya–Vinogradov inequality which takes into account the arithmetic properties of the modulus and we extend the inequality to imprimitive characters. We also find a lower bound for the inequality. 1. introduction Let χ be a nonprincipal Dirichlet character to the modulus q. It has been the M+N interest of mathematicians to study the sum χ(n). Pólya and Vinogradov, n=M+1 independently proved in 1918 that the sum is bounded above by O ( √ q log q). Assuming the Riemann Hypothesis for Lfunctions (GRH), Montgomery [3] showed that the sum is bounded by O ( √ q log log q). This is best possible (up to a constant), because in 1932 Paley [5] proved that there are infinitely many quadratic characters χ such that there exists a constant c> 0 that satisfy for some N the N∑ following inequality χ(n) ∣ ∣> c√q log log q. n=1 Recently, in [2], Levin, Pomerance and Soundararajan considered a “smoothed” version of the Pólya–Vinogradov inequality. Instead of considering the sum of the characters, they consider the weighted sum S ∗ χ(M, N):=
Research Statement
"... My research interests lie in elementary analytic number theory. Most of my work concerns finding explicit estimates for character sums. While these estimates are interesting in their own right, they also are very useful to answer some questions from elementary number theory. For example, I have used ..."
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My research interests lie in elementary analytic number theory. Most of my work concerns finding explicit estimates for character sums. While these estimates are interesting in their own right, they also are very useful to answer some questions from elementary number theory. For example, I have used these estimates to bound the least quadratic nonresidue mod p and to bound the least inert prime in a real quadratic field. Let’s give some background on character sums. For q a positive integer, a Dirichlet character χ mod q is a function χ: N → C that satisfies the following: • For any integer a, χ(a + q) = χ(a), i.e., χ is periodic, with period q. • For any integers a and b, χ(ab) = χ(a)χ(b), i.e., χ is totally multiplicative. • χ(a) = 0 if and only if gcd (a, q)> 1. Dirichlet characters are very important in analytic number theory, an example of an application is the proof that there are infinitely many primes in any arithmetic progression ∞ ∑ χ(n) ax + b as long as gcd(a, b) = 1. This proof depends on the fact that L(1, χ) = n n=1 does not equal 0 for any nonprincipal character χ (principal character mod q means χ(a) = 1 for all integers a such that gcd(a, q) = 1). Let χ be a character mod q and let N, M be integers. Consider Sχ(N, M) = χ(n). M<n≤N+M Notice that if M = 0 then it is the sum of χ evaluated at the first N integers and that by estimating this sum using partial summation, we can estimate L(1, χ). Hence, bounds on Sχ(N, M) give us information on L(1, χ). The first important upper bound on Sχ(N, M) came in 1918 in what we now call the Pólya–Vinogradov inequality (proven independently). The inequality states that there is a universal constant c such that Sχ(N, M)  ≤ c √ q log q for χ a nonprincipal Dirichlet character mod q. Note that, surprisingly, the upper bound does not depend on N, it only depends on the modulus of the character. It is useful to note that χ(n)  = 1 or χ(n) = 0 whenever χ is a Dirichlet character. From this it is trivial to see that Sχ(N, M)  ≤ N. Now if N is small compared to q then the Pólya–Vinogradov inequality is not a good improvement on the trivial bound. Mathematicians have worked out upper bounds for c, for example Pomerance proved the following in [15]: