P. 239 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 23801-23900 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	harmonicbnd4 23801*	The asymptotic behavior of . (Contributed by Mario Carneiro, 14-May-2016.)


Theorem	fsumharmonic 23802*	Bound a finite sum based on the harmonic series, where the "strong" bound only applies asymptotically, and there is a "weak" bound for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

14.3.14 Zeta function

Syntax	czeta 23803	The Riemann zeta function.


Definition	df-zeta 23804*	Define the Riemann zeta function. This definition uses a series expansion of the alternating zeta function ~? zetaalt that is convergent everywhere except , but going from the alternating zeta function to the regular zeta function requires dividing by , which has zeroes other than . To extract the correct value of the zeta function at these points, we extend the divided alternating zeta function by continuity. (Contributed by Mario Carneiro, 18-Jul-2014.)
$\$ $\$

Theorem	zetacvg 23805*	The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)
$/\$

14.3.15 Gamma function

Syntax	clgam 23806	Logarithm of the Gamma function.


Syntax	cgam 23807	The Gamma function.


Syntax	cigam 23808	The inverse Gamma function.
1/

Definition	df-lgam 23809*	Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to because the branch cuts are placed differently (we do have , though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers $\$ , where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.)
$\$ $\$

Definition	df-gam 23810	Define the Gamma function. See df-lgam 23809 for more information about the reason for this definition in terms of the log-gamma function. (Contributed by Mario Carneiro, 12-Jul-2014.)


Definition	df-igam 23811	Define the inverse Gamma function, which is defined everywhere, unlike the Gamma function itself. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/ $\$

Theorem	eldmgm 23812	Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)
$\$ $\$ $/\$

Theorem	dmgmaddn0 23813	If is not a nonpositive integer, then is nonzero for any nonnegative integer . (Contributed by Mario Carneiro, 12-Jul-2014.)
$\$ $\$ $/\$

Theorem	dmlogdmgm 23814	If is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$ $\$

Theorem	rpdmgm 23815	A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	dmgmn0 23816	If is not a nonpositive integer, then is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.)
$\$ $\$

Theorem	dmgmaddnn0 23817	If is not a nonpositive integer and is a nonnegative integer, then is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$ $\$ $\$

Theorem	dmgmdivn0 23818	Lemma for lgamf 23832. (Contributed by Mario Carneiro, 3-Jul-2017.)
$\$ $\$

Theorem	lgamgulmlem1 23819*	Lemma for lgamgulm 23825. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$ $\$ $\$

Theorem	lgamgulmlem2 23820*	Lemma for lgamgulm 23825. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulmlem3 23821*	Lemma for lgamgulm 23825. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulmlem4 23822*	Lemma for lgamgulm 23825. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulmlem5 23823*	Lemma for lgamgulm 23825. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$ $/\$ $/\$

Theorem	lgamgulmlem6 23824*	The series is uniformly convergent on the compact region , which describes a circle of radius with holes of size around the poles of the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$/\$ $/\$

Theorem	lgamgulm 23825*	The series is uniformly convergent on the compact region , which describes a circle of radius with holes of size around the poles of the gamma function. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulm2 23826*	Rewrite the limit of the sequence in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
$/\$ $/\$

Theorem	lgambdd 23827*	The log-Gamma function is bounded on the region . (Contributed by Mario Carneiro, 9-Jul-2017.)
$/\$

Theorem	lgamucov 23828*	The regions used in the proof of lgamgulm 23825 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
$/\$ $\$ $\$ ℂ_fld

Theorem	lgamucov2 23829*	The regions used in the proof of lgamgulm 23825 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
$/\$ $\$ $\$

Theorem	lgamcvglem 23830*	Lemma for lgamf 23832 and lgamcvg 23844. (Contributed by Mario Carneiro, 8-Jul-2017.)
$/\$ $\$ $\$ $/\$

Theorem	lgamcl 23831	The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$

Theorem	lgamf 23832	The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	gamf 23833	The Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	gamcl 23834	The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$

Theorem	eflgam 23835	The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$

Theorem	gamne0 23836	The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	igamval 23837	Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/ $\$

Theorem	igamz 23838	Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ 1/

Theorem	igamgam 23839	Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ $\$ 1/

Theorem	igamlgam 23840	Value of the inverse Gamma function in terms of the log-Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ $\$ 1/

Theorem	igamf 23841	Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theorem	igamcl 23842	Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theorem	gamigam 23843	The Gamma function is the inverse of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ $\$ 1/

Theorem	lgamcvg 23844*	The series converges to . (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	lgamcvg2 23845*	The series converges to . (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	gamcvg 23846*	The pointwise exponential of the series converges to . (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	lgamp1 23847	The functional equation of the (log) Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	gamp1 23848	The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	gamcvg2lem 23849*	Lemma for gamcvg2 23850. (Contributed by Mario Carneiro, 10-Jul-2017.)
$\$ $\$

Theorem	gamcvg2 23850*	An infinite product expression for the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	regamcl 23851	The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	relgamcl 23852	The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	rpgamcl 23853	The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	lgam1 23854	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	gam1 23855	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	facgam 23856	The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	gamfac 23857	The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)


14.4 Basic number theory

14.4.1 Wilson's theorem

Theorem	wilthlem1 23858	The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in are and . (Note that from prmdiveq 14703, is the modular inverse of in . (Contributed by Mario Carneiro, 24-Jan-2015.)
$/\$ $\/$

Theorem	wilthlem2 23859*	Lemma for wilth 23861: induction step. The "hand proof" version of this theorem works by writing out the list of all numbers from to in pairs such that a number is paired with its inverse. Every number has a unique inverse different from itself except and , and so each pair multiplies to , and and multiply to , so the full product is equal to . Here we make this precise by doing the product pair by pair. The induction hypothesis says that every subset of that is closed under inverse (i.e. all pairs are matched up) and contains multiplies to . Given such a set, we take out one element . If there are no such elements, then which forms the base case. Otherwise, $\$ is also closed under inverse and contains , so the induction hypothesis says that this equals ; and the remaining two elements are either equal to each other, in which case wilthlem1 23858 gives that or , and we've already excluded the second case, so the product gives ; or and their product is . In either case the accumulated product is unaffected. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)
mulGrpℂ_fld $/\$ _g _g

Theorem	wilthlem3 23860*	Lemma for wilth 23861. Here we round out the argument of wilthlem2 23859 with the final step of the induction. The induction argument shows that every subset of that is closed under inverse and contains multiplies to , and clearly itself is such a set. Thus, the product of all the elements is , and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)
mulGrpℂ_fld $/\$

Theorem	wilth 23861	Wilson's theorem. A number is prime iff it is greater or equal to and is congruent to , , or alternatively if divides . In this part of the proof we show the relatively simple reverse implication; see wilthlem3 23860 for the forward implication. This is Metamath 100 proof #51. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
$/\$

14.4.2 The Fundamental Theorem of Algebra

Theorem	ftalem1 23862*	Lemma for fta 23869: "growth lemma". There exists some such that is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff deg Poly

Theorem	ftalem2 23863*	Lemma for fta 23869. There exists some such that has magnitude greater than outside the closed ball B(0,r). (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff deg Poly

Theorem	ftalem3 23864*	Lemma for fta 23869. There exists a global minimum of the function . The proof uses a circle of radius where is the value coming from ftalem1 23862; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff deg Poly ℂ_fld

Theorem	ftalem4 23865*	Lemma for fta 23869: Closure of the auxiliary variables for ftalem5 23866. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff deg Poly $/\$ $/\$ $/\$ $/\$

Theorem	ftalem5 23866*	Lemma for fta 23869: Main proof. We have already shifted the minimum found in ftalem3 23864 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let be the lowest term in the polynomial that is nonzero, and let be a -th root of . Then an evaluation of where is a sufficiently small positive number yields for the first term and for the -th term, and all higher terms are bounded because is small. Thus, , in contradiction to our choice of as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff deg Poly

Theorem	ftalem6 23867*	Lemma for fta 23869: Discharge the auxiliary variables in ftalem5 23866. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff deg Poly

Theorem	ftalem7 23868*	Lemma for fta 23869. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff deg Poly

Theorem	fta 23869*	The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. This is Metamath 100 proof #2. (Contributed by Mario Carneiro, 15-Sep-2014.)
Poly $/\$ deg

14.4.3 The Basel problem (ζ(2) = π²/6)

Theorem	basellem1 23870	Lemma for basel 23879. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.) Replace OLD theorem. (Revised ba Wolf Lammen, 18-Sep-2020.)
$/\$

Theorem	basellem2 23871*	Lemma for basel 23879. Show that is a polynomial of degree , and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
Poly $/\$ deg $/\$ coeff

Theorem	basellem3 23872*	Lemma for basel 23879. Using the binomial theorem and de Moivre's formula, we have the identity , so taking imaginary parts yields , where . (Contributed by Mario Carneiro, 30-Jul-2014.)
$/\$

Theorem	basellem4 23873*	Lemma for basel 23879. By basellem3 23872, the expression goes to zero whenever for some , so this function enumerates distinct roots of a degree- polynomial, which must therefore be all the roots by fta1 23129. (Contributed by Mario Carneiro, 28-Jul-2014.)


Theorem	basellem5 23874*	Lemma for basel 23879. Using vieta1 23133, we can calculate the sum of the roots of as the quotient of the top two coefficients, and since the function enumerates the roots, we are left with an equation that sums the function at the different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)


Theorem	basellem6 23875	Lemma for basel 23879. The function goes to zero because it is bounded by . (Contributed by Mario Carneiro, 28-Jul-2014.)


Theorem	basellem7 23876	Lemma for basel 23879. The function for any fixed goes to . (Contributed by Mario Carneiro, 28-Jul-2014.)


Theorem	basellem8 23877*	Lemma for basel 23879. The function of partial sums of the inverse squares is bounded below by and above by , obtained by summing the inequality over the roots of the polynomial , and applying the identity basellem5 23874. (Contributed by Mario Carneiro, 29-Jul-2014.)
$/\$

Theorem	basellem9 23878*	Lemma for basel 23879. Since by basellem8 23877 is bounded by two expressions that tend to , must also go to by the squeeze theorem climsqz 13682. But the series is exactly the partial sums of , so it follows that this is also the value of the infinite sum . (Contributed by Mario Carneiro, 28-Jul-2014.)


Theorem	basel 23879	The sum of the inverse squares is . This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). This is Metamath 100 proof #14. (Contributed by Mario Carneiro, 30-Jul-2014.)


14.4.4 Number-theoretical functions

Syntax	ccht 23880	Extend class notation with the first Chebyshev function.


Syntax	cvma 23881	Extend class notation with the von Mangoldt function.
Λ

Syntax	cchp 23882	Extend class notation with the second Chebyshev function.
ψ

Syntax	cppi 23883	Extend class notation with the prime-counting function pi.
π

Syntax	cmu 23884	Extend class notation with the Möbius function.


Syntax	csgm 23885	Extend class notation with the divisor function.


Definition	df-cht 23886*	Define the first Chebyshev function, which adds up the logarithms of all primes less than . The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)


Definition	df-vma 23887*	Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Definition	df-chp 23888*	Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than . (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ Λ

Definition	df-ppi 23889	Define the prime π function, which counts the number of primes less than or equal to . (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Definition	df-mu 23890*	Define the Möbius function, which is zero for non-squarefree numbers and is or for squarefree numbers according as to the number of prime divisors of the number is even or odd. (Contributed by Mario Carneiro, 22-Sep-2014.)


Definition	df-sgm 23891*	Define the sum of positive divisors function , which is the sum of the xth powers of the positive integer divisors of n. For , counts the number of divisors of , i.e. is the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)


Theorem	efnnfsumcl 23892*	Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
$/\$ $/\$

Theorem	ppisval 23893	The set of primes less than expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)


Theorem	ppisval2 23894	The set of primes less than expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
$/\$

Theorem	ppifi 23895	The set of primes less than is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)


Theorem	sgmss 23896*	The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)


Theorem	prmdvdsfi 23897*	The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)


Theorem	chtf 23898	Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)


Theorem	chtcl 23899	Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)


Theorem	chtval 23900*	Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)

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