Juan Arias de Reyna

A generalized mean-value theorem

We give another proof of Seymour and Zaslavskys theorem: For every familyf1,f2,...,fn of continous functions defined on [0, 1], there exists a finite setF⊂[0, 1] such that the average sum offk overF coincides with the integral offk for everyk=1, 2,...,n.

A PROOF OF A TRIGONOMETRIC INEQUALITY. A GLIMPSE INSIDE THE MATHEMATICAL KITCHEN

We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical technique (maximal slope principle) for proving inequalities between elementary functions. The second proof is fully analytical. Finally we prove a general rearrangement theorem and apply it to the remaining case 0 < r < 1. Some of these inequalities are needed for obtaining general sharp bounds for the errors committed when applying the Riemann-Siegel expansion of Riemanns zeta function.

2. Fourier Series

2.1 Introduction 2.2 Dirichlet Kernel 2.3 Fourier Series of Continuous Functions 2.4 Banach continuity principle 2.5 Summability 2.6 The Conjugate Function 2.7 The Hilbert transform on \({\bf R}\) 2.8 The conjecture of Luzin

Tuples of polynomials over finite fields with pairwise coprimality conditions

Let

On the Sign of the Real Part of the Riemann Zeta Function

q

Some bounds and limits in the theory of Riemann's zeta function

be a prime power. We estimate the number of tuples of degree bounded monic polynomials

9. Pair Interchange Theorems

(Q_1,\ldots,Q_v) \in (\mathbb{F}_q[z])^v

8. Allowed Pairs

that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with finite norms.

7. Carleson analysis of the function

We consider the distribution of the argument of the Riemann zeta function on vertical lines with real part greater than 1/2, and in particular two densities related to the argument and to the real part of the zeta function on such lines. Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function associated with the argument. We give explicit expressions for the densities in terms of this characteristic function. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of the densities.

6. Growth of Partial Sums

Abstract For any real a > 0 we determine the supremum of the real σ such that ζ ( σ + i t ) = a for some real t . For 0 a 1 , a = 1 , and a > 1 the results turn out to be quite different. We also determine the supremum E of the real parts of the ‘turning points’, that is points σ + i t where a curve Im ζ ( σ + i t ) = 0 has a vertical tangent. This supremum E (also considered by Titchmarsh) coincides with the supremum of the real σ such that ζ ′ ( σ + i t ) = 0 for some real t . We find a surprising connection between the three indicated problems: ζ ( s ) = 1 , ζ ′ ( s ) = 0 and turning points of ζ ( s ) . The almost extremal values for these three problems appear to be located at approximately the same height.