4. (the Dirichlet ring) Consider the set of all arithmetic functions, R, with the oper- ations of convolution and point-wise addition, that is, for arithmetic functions f and g define a new arithmetic function (f + g)(n) = f(n) + g(n). This makes R a ring, called the Dirichlet ring. (a) Prove, for example, that f * (g + h) = f * g+ f * h (b) Prove that a f is a unit of R if and only if f(1) +0. (c) Does the subset of multiplicative functions form a subring of R? Justify your an- swer.

4. (the Dirichlet ring) Consider the set of all arithmetic functions, R, with the oper- ations of convolution and point-wise addition, that is, for arithmetic functions f and g define a new arithmetic function (f + g)(n) = f(n) + g(n). This makes R a ring, called the Dirichlet ring. (a) Prove, for example, that f * (g + h) = f * g+ f * h (b) Prove that a f is a unit of R if and only if f(1) +0. (c) Does the subset of multiplicative functions form a subring of R? Justify your an- swer.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

4. (the Dirichlet ring) Consider the set of all arithmetic functions, R, with the oper-
ations of convolution and point-wise addition, that is, for arithmetic functions f and
g define a new arithmetic function (f + g)(n) = f(n) + g(n). This makes R a ring,
called the Dirichlet ring.
(a) Prove, for example, that
f * (g+ h) = f * g+ f * h
(b) Prove that a f is a unit of R if and only if f(1) + 0.
(c) Does the subset of multiplicative functions form a subring of R? Justify your an-
swer.

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