Keith Conrad

Partial Euler Products on the Critical Line

The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Eulerproducts foran elliptic curve L-function at s = 1. Goldfeld laterprovedthat these asymp- totics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of √ 2. We extend Goldfelds theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general √ 2 phenomenonis related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise senseseemsmuch deeperthantheRiemannhypothesis. Overfunctionfields, theEulerproduct asymptotics can sometimes be proved unconditionally.

Irreducible Values of Polynomials: A Non-Analogy

For a polynomial f(T ) ∈ Z[T ], the frequency with which the values f(n) are prime has been considered since at least the 18-th century. Euler observed, in a letter to Goldbach in 1752, that the sequence n + 1 has many prime values for 1 ≤ n ≤ 1500. Legendre assumed an arithmetic progression an+ b with (a, b) = 1 contains infinitely many primes in his work on the quadratic reciprocity law. There is also the old question of twin prime pairs n and n+2, but we will focus here only on a single polynomial (in one variable). An asymptotic estimate for

On Weil's proof of the bound for Kloosterman sums

Abstract While most proofs of the Weil bound on one-variable Kloosterman sums over finite fields are carried out in all characteristics, the original proof of this bound, by Weil, assumes the characteristic is odd. We show how to make Weils argument work in even characteristic, for both ordinary Kloosterman sums and sums twisted by a multiplicative character.

Maclaurin's Inequality and a Generalized Bernoulli Inequality

Summary Maclaurins inequality is a natural, but nontrivial, generalization of the arithmetic-geometric mean inequality. We present a new proof that is based on an analogous generalization of Bernoullis inequality. Applications of Maclaurins inequality to iterative sequences and probability are discussed, along with graph-theoretic versions of the Maclaurin and Bernoulli inequalities.

Hardy-Littlewood Constants

Extending a technique introduced in Golomb’s thesis, we look at a Dirichlet series naturally associated to the Hardy-Littlewood conjecture.

Prime specialization in higher genus I

Abstract A classical conjecture predicts how often a polynomial in takes prime values. The natural analogous conjecture for prime values of a polynomial f(T) ∈ k[u][T], where k is a finite field, is false. The conjecture over k[u] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Möbius function at the values of f on k[u] when f ∈ k[u][Tp ], where p is the characteristic of k. In this paper, for we extend the Möbius periodicity results for k[u]—the affine k-line—to the case when f has coefficients in the coordinate ring A of any higher-genus smooth affine k-curve with one geometric point at infinity. The basic strategy is to pull up results from the genus-0 case by means of well-chosen projections to the affine line. Our techniques can also be used to prove nontrivial properties of a correction factor in the conjecture on primality statistics for values of f ∈ A[Tp ] on A, even as f and k vary.