Topics in analytic number theory

In the first chapter we study the meromorphic continuation of the Dirichlet series [Special characters omitted.] <display-math> <fd> <fl>F<inf>r</inf><fen lp="par">s<rp post="par"></fen>=<sum align="c"> <ll><stk><lyr>m,&ldots;,m<inf>r</inf>&ge;1</lyr><lyr>m<inf>1</inf> &cdots;m<inf>r</inf>&equiv;b<hsp sp="0.265"><fen lp="par"><rm> mod<hsp sp="0.212"><mit>q</mit></rm><rp post="par"></fen></lyr></stk> </ll></sum><hsp sp="0.265"><mit>H<fen lp="par"><g>a</g><rf>log</rf> <fen lp="par">m<inf>1</inf><rm>+&cdots;+<mit>m<inf>r</inf></mit> </rm><rp post="par"></fen><rp post="par"></fen></mit><fr><nu> <g>L</g><fen lp="par">m<inf>1</inf><rp post="par"></fen>&cdots; <g>L</g><fen lp="par"><mit>m<inf>r</inf></mit><rp post="par"></fen> </nu><de><fen lp="par">m<inf>1</inf>+&cdots;+m<inf>r</inf><rp post="par"></fen>^s</de></fr>.</fl> </fd> </display-math> where Λ is the classical Von Mangoldt function, H is a smooth periodic function, b is any integer, and q and r are positive integers. Assuming GRH, we prove that this series is analytic in the half plane [real]( s ) > r - 1⁄2, except for simple poles s = r +2πiαn , n ∈ [Special characters omitted.] <math> <f> <blkbd>Z</blkbd></f> </math> . We also give the residues at these poles. The most important point of this study is that we show these functions have natural boundaries. For example, we show that if we assume GRH, then the line [real](s ) = [Special characters omitted.] <math> <f> <fr><nu>3</nu><de>2</de></fr></f> </math> is the natural boundary of F2 ( s ) under some conditions on H and α. In the second chapter we study the Multiplicative Partition Function. In the third chapter we study an analog of the abc conjecture for arithmetical functions in r variables over a field K of characteristic zero.