4400arithmetic

4400arithmetic - ARITHMETICAL FUNCTIONS I: MULTIPLICATIVE...

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ARITHMETICAL FUNCTIONS I: MULTIPLICATIVE FUNCTIONS PETE L. CLARK 1. Arithmetical Functions Definition: An arithmetical function is a function f : Z + C . Truth be told, this definition is a bit embarrassing. It would mean that taking any function from calculus whose domain contains [1 , + ) and restricting it to positive integer values, we get an arithmetical function. For instance, e - 3 x cos 2 x +(17 log( x +1)) is an arithmetical function according to this definition, although it is, at best, dubious whether this function holds any significance in number theory. If we were honest, the definition we would like to make is that an arithmetical function is a real or complex-valued function defined for positive integer arguments which is of some arithmetic significance , but of course this is not a formal definition at all. Probably it is best to give examples: Example A Ω: The prime counting function n 7→ π ( n ), the number of prime num- bers p , 1 p n . This is the example par excellence of an arithmetical function: approximately half of number theory is devoted to understanding its behavior. This function really deserves a whole unit all to itself, and it will get one: we put it aside for now and consider some other examples. Example 1: The function ω ( n ), which counts the number of distinct prime di- visors of n . Example 2: The function ϕ ( n ), which counts the number of integers k , 1 k n , with gcd( k,n ) = 1. Properly speaking this function is called the totient func- tion , but its fame inevitably precedes it and modern times it is usually called just “the phi function” or “Euler’s phi function.” Since a congruence class k modulo n is invertible in the ring Z /n Z iff its representative k is relatively prime to n , an equivalent definition is ϕ ( n ) := #( Z /n Z ) × , the cardinality of the unit group of the finite ring Z /n Z . Example 3: The function n 7→ d ( n ), the number of positive divisors of n . 1
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2 PETE L. CLARK Example 4: For any integer k , the function σ k ( n ), defined as σ k ( n ) = X d | n d k , the sum of the k th powers of the positive divisors of n . Note that σ 0 ( n ) = d ( n ). Example 5: The M¨obius function μ ( n ), defined as follows: μ (1) = 1, μ ( n ) = 0 if n is not squarefree; μ ( p 1 ··· p r ) = ( - 1) r , when p 1 ,...,p r are distinct primes. Example 6: For a positive integer k , the function r k ( n ) which counts the num- ber of representations of n as a sum of k integral squares: r k ( n ) = # { ( a 1 ,...,a k ) | a 2 1 + ... + a 2 k = n } . These examples already suggest many others. Notably, all our examples but Ex- ample 5 are special cases of the following general construction: if we have on hand, for any positive integer n , a finite set S n of arithmetic objects, then we can define an arithmetic function by defining n 7→ # S n . This shows the link between number theory and combinatorics. In fact the M¨obius function μ is a yet more purely com-
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4400arithmetic - ARITHMETICAL FUNCTIONS I: MULTIPLICATIVE...

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