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Bertelsen's number is an erroneous name erroneously given to the erroneous value of , where is the prime counting function. This value is 56 lower than the correct value of . Ore (1988, p. 69) states that the erroneous value 478 originated in Bertelsen's application of Meissel's method in 1893 (MathPages; Prime Curios!). However, the incorrect value actually first appears in Meissel (1885) rather than Bertelsen in 1893, as correctly noted by Lagarias et al. 1985. (Note that MathPages incorrectly states that Lagarias et al. attribute the result to Bertelsen.)Unfortunately, the incorrect value has continued to be propagated in modern works such as Hardy and Wright (1979, p. 9), Davis and Hersch (1981, p. 175; but actually given correctly in the table on p. 213), Sondheimer (1981), Kramer (1983), Ore (1988, p. 77), and Cormen et al. (1990)...

The Mangoldt function is the function defined by(1)sometimes also called the lambda function. has the explicit representation(2)where denotes the least common multiple. The first few values of for , 2, ..., plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, ... (OEIS A014963).The Mangoldt function is implemented in the WolframLanguage as MangoldtLambda[n].It satisfies the divisor sums(3)(4)(5)(6)where is the Möbius function (Hardy and Wright 1979, p. 254).The Mangoldt function is related to the Riemann zeta function by(7)where (Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).The summatory Mangoldt function, illustratedabove, is defined by(8)where is the Mangoldt function, and is also known as the second Chebyshev function (Edwards 2001, p. 51). is given by the so-called explicit formula(9)for and not a prime or prime power (Edwards 2001, pp. 49, 51, and 53), and the sum is over all nontrivial..

Let be an order of an imaginary quadratic field. The class equation of is the equation , where is the extension field minimal polynomial of over , with the -invariant of . (If has generator , then . The degree of is equal to the class number of the field of fractions of .The polynomial is also called the class equation of (e.g., Cox 1997, p. 293).It is also true thatwhere the product is over representatives of each ideal class of .If has discriminant , then the notation is used. If is not divisible by 3, the constant term of is a perfect cube. The table below lists the first few class equations as well as the corresponding values of , with being generators of ideals in each ideal class of . In each case, the constant term is written out as a cube times a cubefree part.0..

Let be a positive number, and define(1)(2)where the sum extends over the divisors of , and is the Möbius function. Then(3)(Nagell 1951, p. 286).For , 2, ..., is given by 0, 1, 3, 7, 11, 15, 20, 25, ... (OEIS A109507), where is the nearest integer function

The function(1)where is the number of not necessarily distinct prime factors of , with . The values of for , 2, ... are 1, , , 1, , 1, , , 1, 1, , , ... (OEIS A008836). The values of such that are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, ... (OEIS A026424), while then values such that are 1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, ... (OEIS A028260).The Liouville function is implemented in the WolframLanguage as LiouvilleLambda[n].The Liouville function is connected with the Riemannzeta function by the equation(2)(Lehman 1960). It has the Lambert series(3)(4)where is a Jacobi theta function.Consider the summatory function(5)the values of which for , 2, ... are 1, 0, , 0, , 0, , , , 0, , , , , , 0, , , , , ... (OEIS A002819).Lehman (1960) gives the formulas(6)and(7)where , , and are variables ranging over the positive integers, is the Möbius function, is Mertens function, and , , and are positive real numbers with .The conjecture that satisfies for is called the..

The term "left factorial" is sometimes used to refer to the subfactorial , the first few values for , 2, ... are 1, 3, 9, 33, 153, 873, 5913, ... (OEIS A007489).Unfortunately, the same term and notation are also applied to the factorialsum(1)(2)(3)(4)where is a gamma function, is the exponential integral, and is the En-function.For , 1, ..., the first few values are given by 0, 1, 2, 4, 10, 34, 154, 874, ... (OEIS A003422). The left factorial is always even for . is prime for , 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354, 721, 1796, 3733, 4769, 9316, 12221, ... (OEIS A100614), the last of which was found by E. W. Weisstein (Oct. 19, 2006).

If and are relatively prime so that the greatest common divisor , thenwhere is the Carmichael function.

A multiplicative number theoretic function is a number theoretic function that has the property(1)for all pairs of relatively prime positive integers and .If(2)is the prime factorization of a number , then(3)Multiplicative number theoretic functions satisfy the amazing identity(4)(5)where the product is over the primes.

There are two definitions of the Carmichael function. One is the reduced totient function (also called the least universal exponent function), defined as the smallest integer such that for all relatively prime to . The multiplicative order of (mod ) is at most (Ribenboim 1989). The first few values of this function, implemented as CarmichaelLambda[n], are 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, ... (OEIS A002322).It is given by the formula(1)where are primaries.It can be defined recursively as(2)Some special values include(3)and(4)where is a primorial (S. M. Ruiz, pers. comm., Jul. 5, 2009).The second Carmichael's function is given by the least common multiple (LCM) of all the factors of the totient function , except that if , then is a factor instead of . The values of for the first few are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, ... (OEIS A011773).This function has the special value(5)for an odd prime and ...

The so-called explicit formulagives an explicit relation between prime numbers and Riemann zeta function zeros for and not a prime or prime power. Here, is the summatory Mangoldt function (also known as the second Chebyshev function), and the second sum is over all nontrivial zeros of the Riemann zeta function , i.e., those in the critical strip so (Montgomery 2001).

The Möbius function is a number theoretic function defined by(1)so indicates that is squarefree (Havil 2003, p. 208). The first few values of are therefore 1, , , 0, , 1, , 0, 0, 1, , 0, ... (OEIS A008683). Similarly, the first few values of for , 2, ... are 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, ... (OEIS A008966).The function was introduced by Möbius (1832), and the notation was first used by Mertens (1874). However, Gauss considered the Möbius function more than 30 years before Möbius, writing "The sum of all primitive roots [of a prime number ] is either (when is divisible by a square), or (mod ) (when is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)" (Gauss 1801, Pegg 2003).The Möbius function is implemented in the WolframLanguage as MoebiusMu[n].The summatory function of the Möbius function(2)is called the Mertens function.The..

Given relatively prime integers and (i.e., ), the Dedekind sum is defined by(1)where(2)with the floor function. is an odd function since and is periodic with period 1. The Dedekind sum is meaningful even if , so the relatively prime restriction is sometimes dropped (Apostol 1997, p. 72). The symbol is sometimes used instead of (Beck 2000).The Dedekind sum can also be expressed in the form(3)If , let , , ..., denote the remainders in the Euclidean algorithm given by(4)(5)(6)for and . Then(7)(Apostol 1997, pp. 72-73).In general, there is no simple formula for closed-form evaluation of , but some special cases are(8)(9)(Apostol 1997, p. 62). Apostol (1997, p. 73) gives the additional special cases(10)(11)(12)(13)for and , where and . Finally,(14)for and , where or .Dedekind sums obey 2-term(15)(Dedekind 1953; Rademacher and Grosswald 1972; Pommersheim 1993; Apostol 1997, pp. 62-64) and 3-term(16)(Rademacher..

Define(1)where(2)(Davenport 1980, p. 121), is the Mangoldt function, and is the totient function. Now define(3)where the sum is over relatively prime to , , and(4)Bombieri's theorem then says that for fixed ,(5)provided that .

The Mertens function is the summary function(1)where is the Möbius function (Mertens 1897; Havil 2003, p. 208). The first few values are 1, 0, , , , , , , , , , , ... (OEIS A002321). is also given by the determinant of the Redheffer matrix.Values of for , 1, 2, ... are given by 1, , 1, 2, , , 212, 1037, 1928, , ... (OEIS A084237; Deléglise and Rivat 1996).The following table summarizes the first few values of at which for various OEIS such that 13, 19, 20, 30, 33, 43, 44, 45, 47, 48, 49, 50, ...5, 7, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 25, 29, ...3, 4, 6, 10, 15, 16, 22, 26, 27, 28, 35, 36, 38, ...0A0284422, 39, 40, 58, 65, 93, 101, 145, 149, 150, ...1A1186841, 94, 97, 98, 99, 100, 146, 147, 148, 161, ...295, 96, 217, 229, 335, 336, 339, 340, 345, 347, 348, ...3218, 223, 224, 225, 227, 228, 341, 342, 343, 344, 346, ...An analytic formula for is not known, although Titchmarsh (1960) showed that if the Riemann hypothesis holds and if there are no multiple Riemann..

The Dedekind -function is defined by the divisor product(1)where the product is over the distinct prime factors of , with the special case . The first few values are(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)giving 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, ... (OEIS A001615).Sums for include(12)(13)where is the Möbius function.The Dirichlet generating functionis given by(14)(15)where is the Riemann zeta function.

A number is called a barrier of a number-theoretic function if, for all , . Neither the totient function nor the divisor function has a barrier.Let be an open set and , then a function is called a barrier for at a point if 1. is continuous, 2. is subharmonic on , 3. , 4. (Krantz 1999, pp. 100-101).

Given the Mertens function defined by(1)where is the Möbius function, Stieltjes claimed in an 1885 letter to Hermite that stays within two fixed bounds, which he suggested could probably be taken to be (Havil 2003, p. 208). In the same year, Stieltjes (1885) claimed that he had a proof of the general result. However, it seems likely that Stieltjes was mistaken in this claim (Derbyshire 2004, pp. 160-161). Mertens (1897) subsequently published a paper opining based on a calculation of that Stieltjes' claim(2)for was "very probable."The Mertens conjecture has important implications, since the truth of any equalityof the form(3)for any fixed (the form of the Mertens conjecture with ) would imply the Riemann hypothesis. In fact, the statement(4)for any is equivalent to the Riemann hypothesis (Derbyshire 2004, p. 251).Mertens (1897) verified the conjecture for , and this was subsequently extended to by..

Let be an infinite Abelian semigroup with linear order such that is the unit element and implies for . Define a Möbius function on by andfor , 3, .... Further suppose that (the true Möbius function) for all . Then Braun's conjecture states thatfor all .

A class number formula is a finite series giving exactly the class number of a ring. For a ring of quadratic integers, the class number is denoted , where is the discriminant. A class number formula is known for the full ring of cyclotomic integers, as well as for any subring of the cyclotomic integers. This formula includes the quadratic case as well as many cubic and higher-order rings.

The transform inverting the sequence(1)into(2)where the sums are over all possible integers that divide and is the Möbius function.The logarithm of the cyclotomicpolynomial(3)is closely related to the Möbius inversion formula.

Landau (1911) proved that for any fixed ,as , where the sum runs over the nontrivial Riemann zeta function zeros and is the Mangoldt function. Here, "fixed " means that the constant implicit in depends on and, in particular, as approaches a prime or a prime power, the constant becomes large.Landau's formula is roughly the derivative of the explicitformula.Landau's formula is quite extraordinary. If is not a prime or a prime power, then and the sum grows as a constant times . But if is a prime or a prime power, then and the sum grows much faster, like a constant times . This exhibits an amazing connection between the primes and the s; somehow the zeros "recognize" when is a prime and cause large contributions to the sum.

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