By Wolfgang Schwarz

The subject of this e-book is the characterization of sure multiplicative and additive arithmetical services by means of combining equipment from quantity conception with a few basic rules from sensible and harmonic research. The authors accomplish that objective by means of contemplating convolutions of arithmetical services, hassle-free mean-value theorems, and houses of similar multiplicative features. additionally 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 energy sequence with multiplicative coefficients are incorporated, in addition to routines and an intensive bibliography.

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IX(n)I = 1, we obtain a completely multiplicative, m-periodic function X : IN -*{ z E C, Izi = 1 or z = 0 }, defined by X(n) = X(n mod m) If gcd(n,m) = 1, and X(n) = 0 otherwise. 4. THE TURAN-KUBILIUS INEQUALITY An additive function w: IN - C is called strongly additive if the values of w at prime-powers are restricted by the condition w(pk) = w(p), if k = 1, 2, ... 1) 1 ((j(n) - loglog x )2 s c nsx x loglog x with some constant c. P. TURAN used this result to reprove HARDY Tools from Number Theory 20 and RAMANL[IAN's theorem [1917] that ro(n) has normal order loglog n.

In particular, lf(pk)l < if pk 2 N(1). 2. Arithmetical Functions, Convolution, Mobius Inversion Formula 13 some constant y, independent of s, p, k, for which If(pk)I 5 Y. The number of integers, composed entirely from prime-powers pt s N(E), is finite, and so any of these numbers is less than some NP(s). > N(s), which divides n. if n > N*(s), then there is some prime-power Denote by NPP(s) the number of prime-powers below N(E). The function f being multiplicative, we obtain ple I f(n) I s Y1vPP{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. 10) c(s) = 2s r<9-1 sin( its ) F(I-s) c(1-s) of the RIEMANN zeta-function. S. Generating Functions, Dirichlet Series £(f,s) D(g,s) = f(d) nz1 dTn ' 29 g(n/d) ' n-s = D(f*g,s), and so the pointwise product of the DIRICHLET series corresponds to the convolution product of arithmetical functions.

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Arithmetical Functions by Wolfgang Schwarz
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