The Riemann-zeta function on vertical arithmetic progressions

We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression $1/2 + i(an + b)$ with $a > 0$, $b$ real, exhibits a remarkable correspondance with the analogous continuous average and derive several consequences. For example, motivated by the linear independence conjecture, we show at least one third of the elements in the arithmetic progression $a n + b$ are not the ordinates of some zero of $\zeta(s)$ lying on the critical line. This improves on earlier work of Martin and Ng. We then complement this result by producing large and small values of $\zeta(s)$ on arithmetic progressions which are of the same quality as the best $\Omega$ results currently known for $\zeta(1/2 + it)$ with $t$ real.Comment: 20 page

A structure theorem in probabilistic number theory

We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. This is a consequence of a structure theorem making clear the inter-relation between the distribution of an additive function on the integers, and its distribution on the primes.Comment: 10 page

Future EMU Membership and Wage Flexibility in Selected EU Candidate Countries

This paper attempts to evaluate wage rigidity related to risks of increased size and volatility of unemployment after the candidate countries enter the EMU. Such evaluation is done through the study of past labour market adjustment mechanisms and, in particular, the role played by the exchange rate movements and independent monetary policy. The paper examines some institutional and structural characteristics of candidate countries labour markets that could influence the wage elasticity. The analysis indicates that generally nominal wages are not flexible in candidate countries. Inflationary surprises and nominal exchange rate movements have an effect on the adjustment, especially during the Russian crisis. On the other hand fast productivity growth creates the environment in which unit labour can adjust to unfavourable labour market outcomes through moderation of real wage dynamics despite nominal stickiness. The paper indicates possible fields of further in-depth research in this area.labour market, unemployment, European Monetary Union, EU enlargement, EMU enlargement, wage flexibility

Refinements of G\'al's theorem and applications

We give a simple proof of a well-known theorem of G\'al and of the recent related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD sums. In fact, our method obtains the asymptotically sharp constant in G\'al's theorem, which is new. Our approach also gives a transparent explanation of the relationship between the maximal size of the Riemann zeta function on vertical lines and bounds on GCD sums; a point which was previously unclear. Furthermore we obtain sharp bounds on the spectral norm of GCD matrices which settles a question raised in [2]. We use bounds for the spectral norm to show that series formed out of dilates of periodic functions of bounded variation converge almost everywhere if the coefficients of the series are in $L^2 (\log\log 1/L)^{\gamma}$, with $\gamma > 2$. This was previously known with $\gamma >4$, and is known to fail for $\gamma<2$. We also develop a sharp Carleson-Hunt-type theorem for functions of bounded variations which settles another question raised in [1]. Finally we obtain almost sure bounds for partial sums of dilates of periodic functions of bounded variations improving [1]. This implies almost sure bounds for the discrepancy of $\{n_k x\}$ with $n_k$ an arbitrary growing sequences of integers.Comment: 16 page

On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato--Tate density. Examples of such sequences come from coefficients of several $L$-functions of elliptic curves and modular forms. In particular, we show that $|\tau(n)|\le n^{11/2} (\log n)^{-1/2+o(1)}$ for a set of $n$ of asymptotic density 1, where $\tau(n)$ is the Ramanujan $\tau$ function while the standard argument yields $\log 2$ instead of $-1/2$ in the power of the logarithm. Another consequence of our result is that in the number of representations of $n$ by a binary quadratic form one has slightly more than square-root cancellations for almost all integers $n$. In addition we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato--Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally assuming the automorphy of all symmetric powers of the form and seems to be within reach unconditionally using the currently established potential automorphy.Comment: The second version contains some improvements and extensions of previous results, suggested by Maksym Radziwill, who is now a co-autho