This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor (loglogH)−C represent the real truth, and this is indeed proved in part II on the assumption that in the first estimate 2k is an integer. %This is of great interest, for little has been known on the mean value of |ζ(12+it)|k for odd k, say k=1; for even k, see the book by E. C. Titchmarsh [The theory of the Riemann zeta function, Clarendon Press, Oxford, 1951, Theorem 7.19]. The proofs are based on applications of classical function-theoretic theorems, together with mean value theorems for Dirichlet polynomials or series. %In the case of the zeta function, the principle is to write |ζ(s)|k=|ζ(s)k/2|2, where ζ(s)k/2 is related to a rapidly convergent series which is essentially a partial sum of the Dirichlet series of ζ(s)k/2, convergent in the half-plane σ>1.
This paper gives a new elementary proof of the version of Siegel's theorem on L(1,χ)=∑∞n=1χ(n)n−1 for a real character χ(modk). The main result of this paper is the theorem: If 3≤k1≤k2 are integers, χ1(modk1) and χ2(modk2) are two real non-principal characters such that there exists an integer n>0 for which χ1(n)⋅χ2(n)=−1 and, moreover, if L(1,χ1)≤10−40(logk1)−1, then L(1,χ2)>10−4(logk2)−1⋅(logk1)−2k−40000L(1,χ1)2. From this the result of T. Tatuzawa on Siegel's theorem follows.
K. F. Roth proved in 1957 that if 1=q1<q2<q3… is the sequence of square-free integers, then qn+1−qn=O(n313(logn)413). P. Erdös set the problem in more general conditions. If 2≤b1<b2<b3… is a sequence of mutually coprime integers, and ∑1bi<∞, he proved that if 1=q1<q2<q3… is the sequence of integers not divisible by any bi, thenqn+1−qn=O(qθn),
where 0<θ<1. C. Szemeredi made an important progress and proved that if Q(x)=∑qi≤x1, then Q(x+h)−Q(x)>>h, where h≥xθ. Refining some of Szemeredi's ideas, it is proved in this paper that %if 0 < < 1, and ∑1bi<∞, thenQ(x+h)−Q(x)>>h,
where x≥h≥xθ and θ>12 is any constant. %In the later part, using the ideas of Jutila, Brun and I. M. Vinogradov, a stronger version (Theorem 2) is proved.