In this paper, we develop a large sieve type inequality with characters to square moduli. One expects that the result should be weaker than the classical inequality, but, conjecturally at least, not by much. The method is generalizable to higher power moduli.

In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of the form p = am 2 + 1 such that a ≤ p 5/9+ε .

Abstract. -We investigate the first and second moments of shifted convolutions of the generalised divisor function d3(n).

Carbon and alumina are important components in refractories used in the steel industry. High temperature interactions of the refractory materials with liquid metal are crucial in dictating the chemical reactions occurring at the interface, this will control the life of refractory and also the metal quality. Although, the high temperature interactions of graphite and alumina (each separately) with liquid iron have been investigated, fundamental understanding of high temperature interactions of graphite/alumina mixtures with liquid iron is far from complete. This is the focus of the present study. This study reports on the interfacial phenomena during wetting of C/Al 2 O 3 mixtures by liquid iron.On a graphite substrate, the carbon content of liquid iron was 5.62 % at 1 600°C. This is the saturation concentration of carbon under the experimental condition. During the graphite/iron interactions, the contact angle changed from 64°at the initial state, to 38°at the final state. There is good wettability between iron and graphite. However, adding different amounts of alumina in the graphite substrate caused the contact angle to increase. When the alumina in the substrate increased from 16.7 to 23.1 %, the contact angle demonstrated a sharp change from good wetting to almost non-wetting. The underlying phenomena dictating these changes are discussed. The carbon transfer reaction occurring at the solid/iron interface plays a key role in governing the wetting at the interface. At high rates of carbon dissolution, good wetting is observed. However, when the alumina in the substrate increased from 16.7 to 23.1 %, the carbon content of iron shows a sharp decrease, and poor wetting is observed. These results, for the first time, bring out the interdependence of wetting and carbon dissolution during the high temperature interactions of graphite/alumina materials with liquid iron.