By Wolfgang Schwarz

The subject of this publication is the characterization of yes multiplicative and additive arithmetical services by way of combining tools from quantity conception with a few basic principles from useful and harmonic research. The authors do so aim through contemplating convolutions of arithmetical services, simple mean-value theorems, and homes of similar multiplicative features. in addition they turn out the mean-value theorems of Wirsing and Hal?sz and learn the pointwise convergence of the Ramanujan growth. eventually, a few purposes to strength sequence with multiplicative coefficients are incorporated, besides routines and an in depth bibliography.

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**Example text**

Applying the lemma to the DIRICHLET series Vf,s), we obtain the following corollary. 2. 7) Z f(n)-n-' = II ( p nzl 1 f(p2),p-2s + + ... is valid. 8) c(s) = D(1,s) nal n = IfP ( 1+p-s+p-2s+... s =r (i-p-sY1 which is absolutely convergent in Re s > 1. This product representation indicates some connection with the theory of prime numbers. 9) ((s) _ (s - 1) 1 + 2 - s fi Bo(u) u cs+l> du. 9) provides an analytic continuation of C(s) into the half-plane Re s > 0, showing that c(s) has a simple pole at s = 1 with residue 1.

P. [102" c - 102 = b) The formula holds for n = 1, 2, ... 102 ' c . 22) Define the polynomial p(x) by p(X) =lsssn y 1 (x e2 1 ' n) ). 2) In detail. 24) Define D(f) by D(f): n H f(n) log n. Then the map D Is a derivation (so that D: C" '4 is linear, Ds = 0, and D(f*g) = f*D(g) + D(f)*g). 25) g is completely additive if and only if the map f N f g is a derivation. Note that many properties of derivations are dealt with in T. 18. 26) Prove: For every positive integer k, din dk = M + 1) nk' 2: r :1 c r(n) and this series is absolutely convergent.

1. RAMANL(IAN sums have the following properties. (a) The RAMANUJAN sum cr is r-periodic. (b) Cr(n) = 2: dlgcd(r,n) T d (c) The RAMANUJAN sum cr is r-even. (d) For any fixed n the map r H Cr(n) is multiplicative. 3) I cd(m) m=1 c (m) - J l t 0, if d t, cp(d), if d t. Proof. (a) is obvious. 3. periodic Functions, Even Functions, Ramanujan Sums Idir t(d) the latter isasr,a-0 mod d e (r part of the equation above is equal to 17 n); . isbsr/d e( r/d b n and this expression is 0 if (r/d) 4' n, and is equal to r/d otherwise.