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Weierstrass Product Theorem

Let any finite or infinite set of points having no finite limit point be prescribed, and associate with each of its points a definite positive integer as its order. Then there exists an entire function which has zeros to the prescribed orders at precisely the prescribed points, and is otherwise different from zero. Moreover, this function can be represented as a product from which one can read off again the positions and orders of the zeros. Furthermore, if is one such function, thenis the most general function satisfying the conditions of the problem, where denotes an arbitrary entire function.This theorem is also sometimes simply known as Weierstrass's theorem. A spectacularexample is given by the Hadamard product.

Primorial Prime

Primorial primes are primes of the form , where is the primorial of . A coordinated search for such primes is being conducted on PrimeGrid. is prime for , 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... (OEIS A057704; Guy 1994, pp. 7-8; Caldwell 1995). These correspond to with , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, ... (OEIS A006794). The largest known primorial primes as of Nov. 2015 are summarized in the following table (Caldwell).digitsdiscoverer6845Dec. 1992365851PrimeGrid (Dec. 20, 2010)476311PrimeGrid (Mar. 5, 2012) (also known as a Euclid number) is prime for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... (OEIS A014545; Guy 1994, Caldwell 1995, Mudge 1997). These correspond to with , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547,..

Primorial

Let be the th prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by(1)The values of for , 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (OEIS A002110).It is sometimes convenient to define the primorial for values other than just the primes, in which case it is taken to be given by the product of all primes less than or equal to , i.e.,(2)where is the prime counting function. For , 2, ..., the first few values of are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS A034386).The logarithm of is closely related to the Chebyshev function , and a trivial rearrangement of the limit(3)gives(4)(Ruiz 1997; Finch 2003, p. 14; Pruitt), where eis the usual base of the natural logarithm.

Prime Products

The product of primes(1)with the th prime, is called the primorial function, by analogy with the factorial function. Its logarithm is closely related to the Chebyshev function .The zeta-regularized product over allprimes is given by(2)(3)(Muñoz Garcia and Pérez-Marco 2003, 2008), answering the question posed by Soulé et al. (1992, p. 101). A derivation proceeds by algebraic manipulation of the prime zeta function and gives the more general results(4)and(5)(Muñoz Garcia and Pérez-Marco 2003).Mertens theorem states that(6)where is the Euler-Mascheroni constant, and a closely related result is given by(7)There are amazing infinite product formulas forprimes given by(8)(Ramanujan 1913-1914; Le Lionnais 1983, p. 46) and(9)(OEIS A082020; Ramanujan 1913-1914).More general formulas are given by(10)where is the Riemann zeta function and by the Euler product(11)Named prime..

Infinite Cosine Product Integral

At the age of 17, Bernard Mares proposed the definite integral (Borwein and Bailey2003, p. 26; Bailey et al. 2006)(1)(2)(OEIS A091473). Although this is within of ,(3)(OEIS A091494), it is not equal to it. Apparently, no closed-form solution is known for .Interestingly, the integral(4)(5)(Borwein et al. 2004, pp. 101-102) has a value fairly close to , but no other similar relationships seem to hold for other multipliers of the form or .The identity(6)can be expanded to yield(7)In fact,(8)where is a Borwein integral.

Hadamard Product

The Hadamard product is a representation for the Riemann zeta function as a product over its nontrivial zeros ,(1)where is the Euler-Mascheroni constant and is the Gamma function (Titchmarsh 1987, Voros 1987). The constant in the exponent is given by(2)(3)(OEIS A077142). Hadamard used the Weierstrass product theorem to derive this result. The plot above shows the convergence of the formula along the real axis using the first 100 (red), 500 (yellow), 1000 (green), and 2000 (blue) Riemann zeta function zeros.The product can also be stated in the alternate form(4)where is the xi-function and(5)(Havil 2003, p. 204).

Factorial Products

The first few values of (known as a superfactorial) for , 2, ... are given by 1, 2, 12, 288, 34560, 24883200, ... (OEIS A000178).The first few positive integers that can be written as a product of factorials are1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, ... (OEIS A001013).The number of ways that is a product of smaller factorials, each greater than 1, for , 2, ... is given by 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, ... (OEIS A034876), and the numbers of products of factorials not exceeding are 1, 2, 4, 8, 15, 28, 49, 83, ... (OEIS A101976).The only known factorials which are products of factorials in an arithmeticprogression of three or more terms are(1)(2)(3)(Madachy 1979).The only solutions to(4)are(5)(6)(7)(Cucurezeanu and Enkers 1987).There are no nontrivial identities of the form(8)for with for for except(9)(10)(11)(12)(Madachy 1979; Guy 1994, p. 80). Here, "nontrivial" means that identities with , or equivalently are excluded, since..

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