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..
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 quantities obtained from cubic, hexagonal, etc., lattice sums, evaluated at , are called Madelung constants.For cubic lattice sums(1)the Madelung constants expressible in closed form for even indices , a few examples of which are summarized in the following table, where is the Dirichlet beta function and is the Dirichlet eta function.OEISconstant2A0860544A016639To obtain the closed form for , break up the double sum into pieces that do not include ,(2)(3)(4)where the negative sums have been reindexed to run over positive quantities. But , so all the above terms can be combined into(5)The second of these sums can be done analytically as(6)which in the case reduces to(7)The first sum is more difficult, but in the case can be written(8)Combining these then gives the original sum as(9) is given by Benson's formula (Borwein and Bailey 2003, p. 24)(10)(11)(12)(OEIS A085469), where the prime indicates thatsummation over (0, 0, 0)..
An equation for a lattice sum (Borwein and Bailey 2003, p. 26)(1)(2)Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is numerically equal to (OEIS A085469), a value known as "the" Madelung constant.No closed form for is known (Bailey et al. 2006).
Ramanujan's Dirichlet L-series is defined as(1)where is the tau function. Note that the notation is sometimes used instead of (Hardy 1999, p. 164). has properties analogous to the Riemann zeta function, and is implemented as RamanujanTauL[s].Ramanujan conjectured that all nontrivial zeros of lie on the line . satisfies the functional equation(2)(Hardy 1999, p. 173) and has the Euler product representation(3)for (since ) (Apostol 1997, p. 137; Hardy 1999, p. 164). can be split up into(4)where(5)(6)The functions , and are returned by the Wolfram Language commands RamanujanTauTheta[t] and RamanujanTauZ[t], respectively.Ramanujan's tau -function is a real function for real and is analogous to the Riemann-Siegel function . The number of zeros in the critical strip from to is given by(7)where is the Ramanujan theta function. Ramanujan conjectured that the nontrivial zeros of the function are all real.Ramanujan's..
where is a Gaussian sum, and is the gamma function.
A generalization of the Gaussian sum. For and of opposite parity (i.e., one is even and the other is odd), Schaar's identity statesSchaar's identity can also be written so as to be valid for , with even.
The sum(1)where runs through the residues relatively prime to , which is important in the representation of numbers by the sums of squares. If (i.e., and ' are relatively prime), then(2)For argument 1,(3)where is the Möbius function. For general ,(4)where is the totient function.
Cubic lattice sums include the following:(1)(2)(3)where the prime indicates that the origin , , etc. is excluded from the sum (Borwein and Borwein 1986, p. 288).These have closed forms for even ,(4)(5)(6)(7)for , where is the Dirichlet beta function, is the Dirichlet eta function, and is the Riemann zeta function (Zucker 1974, Borwein and Borwein 1987, pp. 288-301). The lattice sums evaluated at are called the Madelung constants. An additional form for is given by(8)for , where is the sum of squares function, i.e., the number of representations of by two squares (Borwein and Borwein 1986, p. 291). Borwein and Borwein (1986) prove that converges (the closed form for above does not apply for ), but its value has not been computed. A number of other related double series can be evaluated analytically.For hexagonal sums, Borwein and Borwein (1987, p. 292) give(9)where . This Madelung constant is expressible in closed..
Kloosterman's sum is defined by(1)where runs through a complete set of residues relatively prime to and is defined by(2)The notation is also used, at least for prime .If (if and are relatively prime), then(3)Kloosterman's sum essentially solves the problem introduced by Ramanujan of representing sufficiently large numbers by quadratic forms . Weil improved on Kloosterman's estimate for Ramanujan's problem with the best possible estimate(4)(Duke 1997).
A Gaussian sum is a sum of the form(1)where and are relatively prime integers. The symbol is sometimes used instead of . Although the restriction to relatively prime integers is often useful, it is not necessary, and Gaussian sums can be written so as to be valid for all integer (Borwein and Borwein 1987, pp. 83 and 86).If , then(2)(Nagell 1951, p. 178). Gauss showed that(3)for odd . Written explicitly(4)(Nagell 1951, p. 177).For and of opposite parity (i.e., one is even and the other is odd), Schaar's identity states(5)Such sums are important in the theory of quadraticresidues.
A serieswhere and are complex and is a monotonic increasing sequence of real numbers. The numbers are called the exponents, and are called the coefficients. When , then , the series is a normal Dirichlet L-series. The Dirichlet series is a special case of the Laplace-Stieltjes transform.
The two functions and defined below are known as the Chebyshev functions.The function is defined by(1)(2)(3)(Hardy and Wright 1979, p. 340), where is the th prime, is the prime counting function, and is the primorial. This function has the limit(4)and the asymptotic behavior(5)(Bach and Shallit 1996; Hardy 1999, p. 28; Havil 2003, p. 184). The notation is also commonly used for this function (Hardy 1999, p. 27).The related function is defined by(6)(7)where is the Mangoldt function (Hardy and Wright 1979, p. 340; Edwards 2001, p. 51). Here, the sum runs over all primes and positive integers such that , and therefore potentially includes some primes multiple times. A simple and beautiful formula for is given by(8)i.e., the logarithm of the least common multiple of the numbers from 1 to (correcting Havil 2003, p. 184). The values of for , 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, ... (OEIS A003418;..
A Lambert series is a series of the form(1)for . Then(2)(3)where(4)The particular case is sometimes denoted(5)(6)(7)for (Borwein and Borwein 1987, pp. 91 and 95), where is a q-polygamma function. Special cases and related sums include(8)(9)(10)(11)(12)(Borwein and Borwein 1997, pp. 91-92), which arise in the reciprocalFibonacci and reciprocal Lucas constants.Some beautiful series of this type include(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)where is the Möbius function, is the totient function, is the number of divisors of , is the q-polygamma function, is the divisor function, is the number of representations of in the form where and are rational integers (Hardy and Wright 1979), is a Jacobi elliptic function (Bailey et al. 2006), is the Liouville function, and is the least significant bit of ...
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..