On A Rapidly Converging Series For The Riemann Zeta Function Alois Pichler Department of Statistics and Operations Research, University of Vienna, Austria, Universitätsstraße 5, 1010 Vienna Abstract To evaluate Riemann’s zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allow to reduce Riemann’s hypothesis to similar properties on polynomials. And interestingly, Laguerre’s polynomials form a kind of leitmotif through all sections. Keywords: Riemann Zeta function, Kummer function, Laguerre polynomials, Fourier transform, Riemann hypothesis. 1. Introduction and Definitions 1.1. Confluent Hypergeometric Functions Confluent hypergeometric functions are special hypergeometric functions, sometimes called also Kummer’s function of first and second kind. They are linear independent solutions of Kum- mer’s differential equation z y00 (z) + (b − z) y0 (z) − a y (z) = 0: The first solution, Kummer’s function of the first kind, is usually given as a globally converging power series − 1 a+i 1 i X i z M (a; b; z) := (1.1) b+i−1 i! arXiv:1201.6538v1 [math.NT] 31 Jan 2012 i=0 i (Kummer’s function of the first kind), and Kummer’s function of the second kind is often given by Γ (1 − b) Γ (b − 1) U (a; b; z) := M (a; b; z) − z1−b M (a − b + 1; 2 − b; z) (1.2) Γ (1 − b + a) Γ (a) Email address: [email protected] (Alois Pichler) Preprint submitted to nowhere April 3, 2019 1 INTRODUCTION AND DEFINITIONS 1.2 Laguerre’s Polynomials Kummer’s transformation states that M (a; b; z) = ez M (b − a; b; −z), which subsequently leads to the identity U (a; b; z) = z1−b U (1 + a − b; 2 − b; z). Another solution of this differential equation – which turns out to be identical to U and thus is simply another representation of U – is obtained as an integral representation by the Laplace transform 1 1 U (a; b; z) = e−ztta−1 (1 + t)b−a−1 dt Γ (a) ˆ0 1−b 1 a−b z −zt t = e a dt: (1.3) Γ (1 + a − b) ˆ0 (1 + t) Some particular and frequently used confluent hypergeometric functions are the upper and lower incomplete Gamma function, which have the representations 1 Γ (s; z) := ts−1e−tdt ˆz = e−zU (1 − s; 1 − s; z) = e−zzsU (1; 1 + s; z) (1.4) and z X (−1)k zs+k γ (s; z) := ts−1e−tdt = ˆ k! s + k 0 k=0 zs zs = M (s; s + 1; −z) = e−z M (1; s + 1; z) : (1.5) s s 1.2. Laguerre’s Polynomials For a a negative integer the defining series for M reduces to a polynomial, which turns out to be closely related to Laguerre’s polynomials: The explicit representation is ! i ! i + α X i + α z j L(α) (z) = M (−i; α + 1; z) = (−1) j : (1.6) i i i − j j! j=0 In view of (1.2) Laguerre’s polynomial may be given by Kummer’s second function as well, (α) (−1)i that is Li (z) = i! U (−i; α + 1; z). This somehow suggests that Laguerre’s polynomial are somewhat in between of both solutions M and U of Kummer’s differential equation. This is central in our investigations and reflected in the results of the next sections. In addition to that it is well-know that Laguerre’s polynomials are orthogonal with respect to the weight-function zαe−z; more precisely we find that 8 1 >0 i , j zαe−zL(α) (z) L(α) (z) dz = < (1.7) ˆ i j > i+α 0 : i Γ (α + 1) i = j; Laguerre’s polynomials thus are orthogonal with respect to the inner product 1 zαe−z hgj f i := g (z) f (z) dz: ˆ0 Γ (α + 1) As a result of the classical theory on Hilbert spaces we may expand a function f in a series p with respect to this orthogonal basis, provided that k f k2 := h f j f i < 1. The function has the 2 2 FOURIER SERIES P (α) (α) (α) expansion f (z) = i=0 fi Li (z) (and is therefore in the closed hull of all Li ), its coefficients taking the explicit form 1 L(α) (z) zαe−z f α = i f (z) dz (1.8) i i+α ˆ0 Γ (α + 1) i (cf. (1.7)). Conversely, the norm can be recovered from the function’s coefficients, as k f k2 = 2 P i+α α 2 i=0 i fi . An elementary example of such a representation in explicit terms is 1 X t i e−tz = L(α) (z) ; (1.9) α+1 i 1 + t (1 + t) i=0 as can be verified straight forward by evaluating the respective integrals for the coefficients (1.8). 1 Notably, this series converges point-wise if t > − 2 (k f k2 < 1), even if α ≤ −1. To give a reference of these classical ingredients aggregated in this section above we would like to refer to the standard work [AS64]. 2. Fourier Series Kummer’s functions allow some explicit representation as series of Laguerre polynomials. Theorem 1 (Expansion of Kummer’s functions in terms of Laguerre polynomials). Suppose that 5 Re (b − a) > 0 and Re (α − 2b) > − 2 , then b−1 −a a X i M (a; b; z) = (−1)i L(β−i) (z) b−β−1 i β−b a i=0 i and −a (1 + α − b)! X i U (a; b; z) = L(α) (z) ; (2.1) (1 + α − b + a)! i b−a−α−2 i=0 i moreover, X L(α) (z) Γ (s; z) = zse−z k (2.2) k+1+α−s k=0 (k + 1) k+1 α 1 for Re s − 2 < 4 . zi i Pi −β (β− j) Proof. As for the proof notice first that i! = (−1) j=0 i− j L j (z), which is a kind of con- verse of (1.6) and verified straight forward by comparing the coefficients of respective powers of z. Substituting this into (1.1), interchanging the order of summation and employing Gauss’ 3 2 FOURIER SERIES hypergeometric theorem (cf. [Koe98]) gives 1 a+i−1 i ! X i X −β M (a; b; z) = (−1)i L(β− j) (z) b+i−1 i − j j i=0 i j=0 1 1 a+i−1 ! X X i −β = L(β− j) (z) (−1)i j b+i−1 i − j j=0 i= j i X Γ (b) Γ (b − a − β) Γ (a + j) = L(β− j) (z)(−1) j j Γ (a) Γ (b − a) Γ (b − β + j) j=0 b−1 a+ j−1 X a j = L(β− j) (z)(−1) j j b−β−1 b−β−1+ j j=0 a j b−1 −a a X j = (−1) j L(β− j) (z) ; b−β−1 j β−b a j=0 j which is the desired assertion. As regards convergence it follows from the elementary recursion ! z − β − 1 z L(β− j) (z) = − 1 + L(β− j+1) (z) − L(β− j+2) (z) (2.3) j j j−1 j j−2 L(β− j)(z) j (β− j) −β−1 j z 1 that (−1) L j (z) = O j , a more thorough analysis even discloses that β = e 1 + O j . ( j) −a ( ) a−1 − j (β− j) j O −β−1 j O 1 Hence, ( 1) L j (z) β−b = j jb−β−1 = jb−a+1 , and the series converges – irrespective ( j ) of β – if Re (b − a) > 0. To verify the second statement notice that 1 L(α) (z) zαe−z Uα = i U (a; b; z) dz i i+α ˆ0 Γ (α + 1) i (α) 1 L (z) zαe−z 1 1 = i e−ztta−1 (1 + t)b−a−1 dtdz i+α ˆ0 Γ (α + 1) Γ (a) ˆ0 i 1 ta−1 1 L(α) (z) zαe−z = (1 + t)b−a−1 i e−ztdzdt i+α ˆ0 Γ (a) ˆ0 Γ (α + 1) i 1 1 ti ta−1 ( t)b−a−1 t; = 1 + α+i+1 d Γ (a) ˆ0 (1 + t) which is a consequence of (1.3) and (1.9). The latter integral is a beta function, we thus continue and find the coefficient 4 2 FOURIER SERIES (α) (α) α−1−z (α) α−1 (α) Algorithm 1 to evaluate Li (z), based on Li (z) = 2 + i Li−1 (z) − 1 + i Li−2 (z). L1 : = 0; Laguerre:= 1 For j := 1 t o i L0:= L1 ; L1:= L a g u e r r e ; L a g u e r r e := ( ( 2 ∗ j + alpha − 1− z ) ∗ L1− ( j + alpha − 1)∗ L0 ) / j ; Next j Return Laguerre (β−i) Algorithm 2 to evaluate Li (z), based on (2.3) L1 : = 0; Laguerre:= 1 For j := 1 t o i L0:= L1 ; L1:= L a g u e r r e ; L a g u e r r e := ( ( b e t a+ 1− j − z ) ∗ L1− z ∗ L0 ) / j ; Next j Return Laguerre 1 α 1 a−1+i b−a−α−i−2 Ui = t (1 + t) dt Γ (a) ˆ0 1 Γ (a + i) Γ (α + 2 − b) = Γ (a) Γ (i + a + α + 2 − b) a+i−1 (α + 1 − b)! i = (i + a + α + 1 − b)! i+a+α+2−b i which is the respective coefficient for U.