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Weber Functions

Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as(1)These function may also be written as(2)where is a regularized hypergeometric function.This function is implemented in the Wolfram Language as WeberE[nu, z] and is an analog of the Anger function.Special values for real include(3)(4)(5)(6)where is a Struve function.Letting be a root of unity, another set of Weber functions is defined as(7)(8)(9)(10)(11)(Weber 1902, Atkin and Morain 1993), where is the Dedekind eta function and is the half-period ratio. These functions are related to the Ramanujan g- and G-functions and the elliptic lambda function.The Weber functions satisfy the identities(12)(13)(14)(15)(16)(17)(Weber 1902, Atkin and Morain 1993)...

Artin's Conjecture

There are at least two statements which go by the name of Artin's conjecture.If is any complex finite-dimensional representation of the absolute Galois group of a number field, then Artin showed how to associate an -series with it. These -series directly generalize zeta functions and Dirichlet -series, and as a result of work by Richard Brauer, is known to extend to a meromorphic function on the complex plane. Artin's conjecture predicts that it is in fact holomorphic, i.e., has no poles, with the possible exception of a pole at (Artin 1923/1924). Compare with the generalized Riemann hypothesis, which deals with the locations of the zeros of certain -series.The second conjecture states that every integer not equal to or a square number is a primitive root modulo for infinitely many and proposes a density for the set of such which are always rational multiples of a constant known as Artin's constant. There is an analogous theorem for functions instead..

Riemann Hypothesis

First published in Riemann's groundbreaking 1859 paper (Riemann 1859), the Riemann hypothesis is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the values of other than , , , ... such that (where is the Riemann zeta function) all lie on the "critical line" (where denotes the real part of ).A more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.Legend holds that the copy of Riemann's collected works found in Hurwitz's library after his death would automatically fall open to the page on which the Riemann hypothesis was stated (Edwards 2001, p. ix).While it was long believed that Riemann's hypothesis was the result of deep intuition on the part of Riemann, an examination of his papers by C. L. Siegel showed that Riemann had made detailed..

Gram Point

Let be the Riemann-Siegel function. The unique value such that(1)where , 1, ... is then known as a Gram point (Edwards 2001, pp. 125-126).An excellent approximation for Gram point can be obtained by using the first few terms in the asymptotic expansion for and inverting to obtain(2)where is the Lambert W-function. This approximation gives as error of for , decreasing to by .The following table gives the first few Gram points.OEIS0A11485717.84559954041A11485823.1702827012227.6701822178331.7179799547435.4671842971538.9992099640642.3635503920745.5930289815848.7107766217951.73384281331054.6752374468The integers closest to these points are 18, 23, 28, 32, 35, 39, 42, 46, 49, 52,55, 58, ... (OEIS A002505).There is a unique point at which , given by the solution to the equation(3)and having numerical value(4)(OEIS A114893).It is usually the case that . Values of for which this does not hold are , 134, 195, 211, 232, 254,..

Montgomery's Pair Correlation Conjecture

Montgomery's pair correlation conjecture, published in 1973, asserts that the two-point correlation function for the zeros of the Riemann zeta function on the critical line isAs first noted by Dyson, this is precisely the form expected for the pair correlation of random Hermitian matrices (Derbyshire 2004, pp. 287-291).

Zeta Function

A function that can be defined as a Dirichlet series, i.e., is computed as an infinite sum of powers,where can be interpreted as the set of zeros of some function. The most commonly encountered zeta function is the Riemann zeta function,

Prime Zeta Function

The prime zeta function(1)where the sum is taken over primes is a generalizationof the Riemann zeta function(2)where the sum is over all positive integers. In other words, the prime zeta function is the Dirichlet generating function of the characteristic function of the primes . is illustrated above on positive the real axis, where the imaginary part is indicated in yellow and the real part in red. (The sign difference in the imaginary part compared to the plot appearing in Fröberg is presumably a result of the use of a different convention for .)Various terms and notations are used for this function. The term "prime zeta function" and notation were used by Fröberg (1968), whereas Cohen (2000) uses the notation .The series converges absolutely for , where , can be analytically continued to the strip (Fröberg 1968), but not beyond the line (Landau and Walfisz 1920, Fröberg 1968) due to the clustering of singular..

Gram's Law

Gram's law (Hutchinson 1925; Edwards 2001, pp. 125, 127, and 171) is the tendency for zeros of the Riemann-Siegel function to alternate with Gram points. Stated more precisely, it notes the tendency for to hold, where is a Gram point.Strictly speaking, the statement "" should perhaps be called the weak Gram's law since Hutchinson (1925) used the term "Gram's law" to refer to the stronger statement that there are precisely zeros of between 0 and (Edwards 2001, p. 171).

Nint Zeta Function

Let(1)where denotes nearest integer function, i.e., the integer closest to . For ,(2)(3)(4) is a polynomial in whose coefficients are algebraic numbers whenever is odd. The first few values are given explicitly by(5)(6)(7)

Multivariate Zeta Function

Multivariate zeta function, also called multiple zeta values, multivariate zeta constants (Bailey et al. 2006, p. 43), multi-zeta values (Bailey et al. 2006, p. 17), and multivariate zeta values, are defined by(1)(Broadhurst 1996, 1998). This can be written in the more compact and convenient form(2)(Broadhurst 1996; Bailey et al. 2007, p. 38).The notation (as opposed to ) is sometimes also used to indicate that a factor of 1 in the numerator is replaced by a corresponding factor of . In addition, the notation is used in quantum field theory.In particular, for , these correspond to the usual Euler sums(3)(4)(5)(6)(Broadhurst 1996).Multivariate zeta functions (and their derivatives) also arise in the closed-form evaluation of definite integrals involving the log cosine function (Oloa 2011).These sums satisfy(7)for , as well as(8)for nonnegative integers and (Bailey et al. 2007). These give the special cases(9)(10)(11)(Bailey..

Euler Product

For , the Riemann zeta function is given by(1)(2)where is the th prime. This is Euler's product (Whittaker and Watson 1990), called by Havil (2003, p. 61) the "all-important formula" and by Derbyshire (2004, pp. 104-106) the "golden key."This can be proved by expanding the product, writing each term as a geometricseries, expanding, multiplying, and rearranging terms,(3)Here, the rearrangement leading to equation (1) follows from the fundamental theorem of arithmetic, since each product of prime powers appears in exactly one denominator and each positive integer equals exactly one product of prime powers.This product is related to the Möbius function via(4)which can be seen by expanding the product to obtain(5)(6)(7)(8)(9), but the finite product exists, giving(10)For upper limits , 1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, ... (OEIS A060753 and..

Voronin Universality Theorem

Voronin (1975) proved the remarkable analytical property of the Riemann zeta function that, roughly speaking, any nonvanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the zeta function in the critical strip.More precisely, let and suppose that is a nonvanishing continuous function on the disk that is analytic in the interior. Then for any , there exists a positive real number such thatMoreover, the set of these has positive lower density, i.e.,Garunkštis (2003) obtained explicit estimates for the first approximating and the positive lower density, provided that is sufficiently small and sufficiently smooth. The condition that have no zeros for is necessary.The Riemann hypothesis is known to be true iff can approximate itself uniformly in the sense of Voronin's theorem (Bohr 1922, Bagchi 1987). It is also known that there exists a rich zoo of Dirichlet series having this or some similar..

Dirichlet Lambda Function

Min Max Re Im The Dirichlet lambda function is the Dirichlet L-series defined by(1)(2)where is the Riemann zeta function. The function is undefined at . It can be computed in closed form where can, that is for even positive .The Dirichlet lambda function is implemented in the WolframLanguage as DirichletLambda[x].It is related to the Riemann zeta functionand Dirichlet eta function by(3)and(4)(Spanier and Oldham 1987). Special values of include(5)(6)

Critical Strip

The region , where is defined as the real part of a complex number . All nontrivial zeros (i.e., those not at negative even integers) of the Riemann zeta function lie inside this strip.

Lindelöf Hypothesis

Let be the least upper bound of the numbers such that is bounded as , where is the Riemann zeta function. Then the Lindelöf hypothesis states that is the simplest function that is zero for and for .The Lindelöf hypothesis is equivalent to the hypothesis that (Edwards 2001, p. 186).Backlund (1918-1919) proved that the Lindelöf hypothesis is equivalent to the statement that for every , the number of roots in the rectangle grows less rapidly than as (Edwards 2001, p. 188).

Critical Line

The line in the complex plane on which the Riemann hypothesis asserts that all nontrivial (complex) Riemann zeta function zeros lie. The plot above shows the first few zeros of the Riemann zeta function, with the critical line shown in red. The zeros with and that do not line on the critical line are the trivial zeros of at , , .... Although it is known that an infinite number of zeros lie on the critical line and that these comprise at least 40% of all zeros, the Riemann hypothesis is still unproven.An attractive poster plotting the Riemann zeta function zeros on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).In the Season 1 episode "Prime Suspect" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes discusses the critical line after realizing that character Ethan's daughter has been kidnapped because he is close to solving..

Li's Criterion

Li's criterion states that the Riemann hypothesisis equivalent to the statement that, for(1)where is the xi-function, for every positive integer (Li 1997). Li's constants can be written in alternate form as(2)(Coffey 2004). can also be written as a sum of nontrivial zeros of as(3)(Li 1997, Coffey 2004).A recurrence for in terms of is given by(4)(Coffey 2004).The first few explicit values of the constantes are(5)(6)(7)where is the Euler-Mascheroni constant and are Stieltjes constants. can be computed efficiently in closed form using recurrence formulas due to Coffey (2004), namely(8)where(9)and .OEIS10.0230957...A07476020.0923457...A10453930.2076389...A10454040.3687904...A10454160.5755427...A10454271.1244601...A30634081.4657556...A306341Edwards 2001 (p. 160) gave a numerical value for , and numerical values to six digits up to were tabulated by Coffey (2004).While the values of up to are remarkably well..

Berry Conjecture

The longstanding conjecture that the nonimaginary solutions of(1)where is the Riemann zeta function, are the eigenvalues of an "appropriate" Hermitian operator . Berry and Keating (1999) further conjecture that this operator is(2)(3)where and are the position and conjugate momentum operators, respectively, and multiplication is noncommutative. Note that is symmetric but might have nontrivial deficiency indices, so while physicists define this operator to be Hermitian, mathematicians do not.

Riemann Zeta Function

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is denoted and is plotted above (using two different scales) along the real axis. Min Max Re Im In general, is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). is implemented in the Wolfram Language as Zeta[s].The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not..

Lehmer's Phenomenon

The appearance of nontrivial zeros (i.e., those along the critical strip with ) of the Riemann zeta function very close together. An example is the pair of zeros given by and , illustrated above in the plot of . This corresponds to the region near Gram point (Lehmer 1956; Edwards 2001, p. 178).Let be the th nontrivial root of , and consider the local extrema of . Then the values of after which the absolute value of the local extremum between and decreases are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, ... (OEIS A114886).

Landau's 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.

Hurwitz Zeta Function

The Hurwitz zeta function is a generalization of the Riemann zeta function that is also known as the generalized zeta function. It is classically defined by the formula(1)for and by analytic continuation to other , where any term with is excluded. It is implemented in this form in the Wolfram Language as HurwitzZeta[s, a].The slightly different form(2)is implemented in the Wolfram Language as Zeta[s, a]. Note that the two are identical only for .The plot above shows for real and , with the zero contour indicated in black.For , a globally convergent series for (which, for fixed , gives an analytic continuation of to the entire complex -plane except the point ) is given by(3)(Hasse 1930).The Hurwitz zeta function is implemented in the Wolfram Language as Zeta[s, a].For , reduces to the Riemann zeta function ,(4)If the singular term is excluded from the sum definition of , then as well.The Hurwitz zeta function is given by the integral(5)for and .The..

Hurwitz's Formula

There are a number of formulas variously known as Hurwitz's formula.The first iswhere is a Hurwitz zeta function, is the gamma function, and is the periodic zeta function (Apostol 1995; 1997, p. 71).Hurwitz has another formula, also known as Hurwitz's theorem or the Riemann-Hurwitz formula. Let and be compact Riemann surfaces, and suppose that there is a non-constant analytic map . The Hurwitz formula gives the relationship between the genus of and the genus of , namely,In this formula, is the degree of the map. The degree of is an integer such that for a generic point , (i.e., for all but finitely many points in ), the set consists of points in . The sum in the Hurwitz formula can be viewed as a correction term to take into account the points where . Such points are sometimes called branch points. The numbers are the ramification indices.Hurwitz's theorem for Riemann surfaces essentially follows from an application of the polyhedral formula...

Alternating Harmonic Series

The alternating harmonic series is the serieswhich is the special case of the Dirichlet eta function and also the case of the Mercator series.

Riemann Zeta Function zeta(2)

The value for(1)can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970, Kimble 1987, Knopp and Schur 1918, Kortram 1996, Matsuoka 1961, Papadimitriou 1973, Simmons 1992, Stark 1969, 1970, Yaglom and Yaglom 1987). is therefore the definite sum version of the indefinite sum(2)(3)where is a generalized harmonic number (whose numerator is known as a Wolstenholme number) and is a polygamma function.The problem of finding this value analytically is sometimes known as the Basel problem (Derbyshire 2004, pp. 63 and 370) or Basler problem (Castellanos 1988). It was first proposed by Pietro Mengoli in 1644 (Derbyshire 2004, p. 370). The solution(4)was first found by Euler in 1735 (Derbyshire 2004, p. 64) or 1736 (Srivastava 2000).Yaglom and Yaglom (1987), Holme (1970), and Papadimitriou (1973) all derive the result, from de Moivre's identity or related identities. is given by the..

Dirichlet Eta Function

Min Max Min Max Re Im The Dirichlet eta function is the function defined by(1)(2)where is the Riemann zeta function. Note that Borwein and Borwein (1987, p. 289) use the notation instead of . The function is also known as the alternating zeta function and denoted (Sondow 2003, 2005). is defined by setting in the right-hand side of (2), while (sometimes called the alternating harmonic series) is defined using the left-hand side. The function vanishes at each zero of except (Sondow 2003).The eta function is related to the Riemann zeta function and Dirichlet lambda function by(3)and(4)(Spanier and Oldham 1987). The eta function is also a special case of the polylogarithmfunction,(5)The value may be computed by noting that the Maclaurin series for for is(6)Therefore, the natural logarithm of 2 is(7)(8)(9)(10)The derivative of the eta function is given by(11)or in the special case , by(12)(13)(14)(15)This latter fact provides a remarkable..

Dirichlet Beta Function

Min Max Min Max Re Im The Dirichlet beta function is defined by the sum(1)(2)where is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function by(3)The beta function can be defined over the whole complexplane using analytic continuation,(4)where is the gamma function.The Dirichlet beta function is implemented in the WolframLanguage as DirichletBeta[x].The beta function can be evaluated directly special forms of arguments as(5)(6)(7)where is an Euler number.Particular values for are(8)(9)(10)(11)where is Catalan's constant and is the polygamma function. For , 3, 5, ..., , where the multiples are 1/4, 1/32, 5/1536, 61/184320, ... (OEIS A046976 and A053005).It is involved in the integral(12)(Guillera and Sondow 2005).Rivoal and Zudilin (2003) proved that at least one of the seven numbers , , , , , , and is irrational.The derivative can also be computed analytically at a number of integer values of including(13)(14)(15)(16)(17)(18)(19)(OEIS..

Epstein Zeta Function

The Epstein zeta function for a matrix of a positive definite real quadratic form and a complex variable with (where denotes the real part) is defined by(1)where the sum is over all column vectors with integer coordinates and the prime means the summation excludes the origin (Terras 1973). Epstein (1903) derived the analytic continuation, functional equation, and so-called Kronecker limit formula for this function.Epstein (1903) defined this function in the course of an effort to find the most general possible function satisfying a functional equation similar to that satisfied by the Riemann zeta function (Glasser and Zucker 1980, p. 68).A slightly different notation is used in theoretical chemistry, where the Epstein zeta function arises in connection with lattice sums. Let be a positive definite quadratic form(2)where with , ... is a symmetric matrix. Then the Epstein zeta function can be defined as(3)where and are arbitrary..

Robin's Theorem

Consider the inequalityfor integer , where is the divisor function and is the Euler-Mascheroni constant. This holds for 7, 11, 13, 14, 15, 17, 19, ... (OEIS A091901), and is false for 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, and 5040 (OEIS A067698).Robin's theorem states that the truth of the inequality for all is equivalent to the Riemann hypothesis (Robin 1984; Havil 2003, p. 207).

Riemann Zeta Function Zeros

Zeros of the Riemann zeta function come in two different types. So-called "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" occur at certain values of satisfying(1)for in the "critical strip" . In general, a nontrivial zero of is denoted , and the th nontrivial zero with is commonly denoted (Brent 1979; Edwards 2001, p. 43), with the corresponding value of being called .Wiener (1951) showed that the prime number theorem is literally equivalent to the assertion that has no zeros on (Hardy 1999, p. 34; Havil 2003, p. 195). The Riemann hypothesis asserts that the nontrivial zeros of all have real part , a line called the "critical line." This is known to be true for the first zeros.An attractive poster plotting zeros of the Riemann zeta function on the critical line together with annotations for relevant historical information, illustrated above,..

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