SUMS OF KLOOSTERMAN SUMS OVER PRIMES IN AN ARITHMETIC PROGRESSION

Trigonometric sums over primes III

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Arithmetic-progression-weighted Subsequence Sums

Let G be an abelian group, let S be a sequence of terms s1, s2, . . . , sn ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let W ̄ S = {w1s1 + . . . + wnsn : wi a term of W, wi 6= wj for i 6= j}, which is a particular kind of weighted restricted sumset. We show that |W ̄S| ≥ min{|G| − 1, n}, that W ̄ S = G if n ≥ |G| + 1, and also c...

Legendre Sums, Soto-andrade Sums and Kloosterman Sums

The three sums named in the title are all known to appear in connection with the complex representation theory of GL(2, q). The first two are incarnations of certain spherical vectors, whereas the third is a matrix coefficient for a parabolic basis. In this work, Legendre and Soto-Andrade sums are shown to occur in a second way, as parabolic ClebschGordan coefficients for the tensor product of ...

On zeros of Kloosterman sums

Vinogradov’s estimate for exponential sums over primes

1Many of the manipulations of sums in these chapters are hard to follow, and I greatly expand on the calculations in Davenport. The organization of the proof in Davenport seems to be due to Vaughan. I have also used lecture notes by Andreas Strömbergsson, http://www2.math.uu.se/~astrombe/analtalt08/www_notes.pdf, pp. 245–257. Another set of notes, which I have not used, are http://jonismathnote...