Analytic continuation of modified L-function
By a generalized system of prime numbers (g-primes) a sequence \(q_ n\) is meant which satisfies the conditions \(1<q_ 1\leq q_ 2\leq...\), \(q_ n\to \infty\). The multiplicative semigroup generated by the sequence \(\{q_ n\}\) is the set \(1=n_ 1<n_ 2\leq n_ 3\leq..\). which is called a sequence of g-integers. \textit{C. Ryavec} [Ill. J. Math. 17, 608-618 (1973; Zbl 0266.10042)] studied the class of g-primes of the form \(\{\) vp\(\}\), where p are rational primes and \(v>1/2\). There it was proved that the modified zeta-function \[ \zeta (s,v):=\prod_{p}(1-(vp)^{- s})^{-1}=\sum^{\infty}_{j=1}n_ j^{-s},\quad Re(s)>1 \] can be continued analytically to the half-plane \(Re(s)>0\) outside of a ''comb- like'' set which depends on the zeros of the Riemann zeta-function. In this note a more general situation is considered: Let \(\chi\) be a Dirichlet character mod k, v a natural number. Then the author studies the function \[ L(s,v,\chi):=\prod_{p}(1-\chi(vp)^{-s})^{-1} = \sum^{\infty}_{j=1}\chi(n_ j) n_ j^{-s}. \] Here the product over all rational primes converges for \(Re(s)>1\). It is proved by the method of Ryavec that the series \(L(s,v,\chi)\) can also be continued analytically to the half-plane with ''slits'' with respect to the zeros of the Dirichlet L-functions. Then an asymptotic formula is obtained for the number of g-integers \(n_ j\), \(n_ j\equiv \ell (mod k)\) with \((k,v)=(1,k)=1\), not exceeding x. Because the author gets the sharp remainder term \(O(x \exp(-c\sqrt{\log x})\), \(c>0\) he has studied carefully the dependence of the remainder term from the involved moduli, including Siegel zeros.
scientific article; zbMATH DE number 3902763
scientific article