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Theoretic Arithmetic of the Pythagoreans - Thomas Taylor - Google книги
The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Equivalence of Lie theoretic definitions of an arithmetic lattice Ask Question. Asked 7 months ago. These important functions which are not arithmetic functions are defined for non-negative real arguments, and are used in the various statements and proofs of the prime number theorem.
They are summation functions see the main section just below of arithmetic functions which are neither multiplicative nor additive. It is the summation function of the characteristic function of the prime numbers. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integers modulo n.
Theoretic Arithmetic by Thomas Taylor, A. J. Valpy
The notation is ambiguous, as there are in general many extensions with the same discriminant. See quadratic field and cyclotomic field for classical examples. Given an arithmetic function a n , its summation function A x is defined by. A can be regarded as a function of a real variable. Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right:.
Individual values of arithmetic functions may fluctuate wildly — as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function for large x. A classical example of this phenomenon  is given by the divisor summatory function , the summation function of d n , the number of divisors of n :.
An average order of an arithmetic function is some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average". We say that g is an average order of f if. The example above shows that d n has the average order log n. Given an arithmetic function a n , let F a s , for complex s , be the function defined by the corresponding Dirichlet series where it converges : .
F a s is called a generating function of a n. Consider two arithmetic functions a and b and their respective generating functions F a s and F b s. The product F a s F b s can be computed as follows:.
If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative. There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions.
The page divisor sum identities contains many more generalized and related examples of identities involving arithmetic functions.
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There is a formula for r 3 in the section on class numbers below. Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series :. See Eisenstein series for a discussion of the series and functional identities involved in these formulas.