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Fourier series for zeta function via Sinc
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Linear Algebra and its Applications 429 (2008) 2636–2639 www.elsevier.com/locate/laa
Fourier series for zeta function via Sinc Frank Stenger School of Computing, University of Utah, Salt Lake City, UT 84112, USA Received 30 June 2006; accepted 16 January 2008 Available online 11 March 2008 Submitted by A. Frommer This paper is presented in honor of Richard Varga’s 80th birthday
Abstract In this paper we derive some Fourier series and Fourier polynomial approximations to a function F which has the same zeros as the zeta function, ζ (z) on the strip {z ∈ C : 0 < z < 1}. These approximations depend on an arbritrary positive parameter h, and which for arbitrary ε ∈ (0, 1/2), converge uniformly to ζ (z) on the rectangle {z ∈ C : ε < z < 1 − ε, −π/ h < z < π/ h}. © 2008 Elsevier Inc. All rights reserved. Keywords: Riemann zeta function; Zeros; Fourier series; Sinc methods
1. Results This paper is presented in honor of Richard Varga, who has made important contributions to the determination of the moments of the Riemann ζ -function and the de Bruijn–Newman constant [1, Chapter 3]. The usual series expansion of the zeta function can be obtained via term-wise integration of the expression ∞ 1 nz n=1 ∞ z−1 t 1 = dt (z) 0 et − 1
ζ (z) =
E-mail address: [email protected] 0024-3795/$ - see front matter ( 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2008.01.037
(1.1)
F. Stenger / Linear Algebra and its Applications 429 (2008) 2636–2639
=
1 (z)
∞
2637
e−t t z−1 (1 + e−t + e−2t + · · ·)dt,
0
where (z) denotes the usual gamma function. Unfortunately, the above infinite series converges only in the region z > 1, and it is thus not convenient for studying the zeros of the zeta function in the critical strip, D = {z ∈ C : 0 < z < 1}. However, the above series for ζ (z) immediately yields ∞
2 2 ζ (z) = . z 2 (2n)z
(1.2)
n=1
By subtracting the series (1.2) from the series in (1.1), and then proceeding as in (1.1) above, we immediately get ∞ z−1 t 2 dt, (1.3) F (z) ≡ 1 − z (z) ζ (z) = 2 et + 1 0 and the integral representation in this expression converges to F (z) in the region {z ∈ C : z > 0}. Note also, that the factor multiplying ζ (z) in (1.3) does not have any zeros in the critical strip, D, so that both of the functions F (z) and ζ (z) have the same zeros in D. 2. Application of Sinc quadrature The conditions for convergence of Sinc quadrature are quite simple [2, Section 4.2]. Assume that the function g in the integral ∞ I= g(t)dt. (2.1) 0
satisfies the following assumptions: 1. g is analytic in the sector, Sε ≡ {t ∈ C : | arg(t)| < π/2 − ε}, with ε an arbitrary positive constant in the interval (0, 1), and 2. g ≡ ∂Sε |g(t)dt| < ∞. Then, for arbitrary bounded h > 0, we have [2, Theorem 4.2.2 (b)] ∞ 2π(π/2 − ε) kh kh e g e < C exp − , I − h h
(2.2)
k=−∞
where C is a constant that is proportional to g. Now, by taking g(t) = t z−1 /(exp(t) + 1), we get a function g which not only depends on t but also on a complex parameter z. Furthermore, if z ∈ Dε = {z ∈ C : ε < z < 1 − ε}, (where we assume that 0 < ε < 1/2) then this function g does, in fact, satisfy all of the above conditions, enabling us to arrive at the following result. Theorem 2.1. Let F be defined by the integral (1.3), let z = x + iy ∈ Dε , and set ⎧ ∞ ⎨ exp(nhx) eikhy if |y| < π/ h, Fh (x + iy) = k=−∞ exp(enh )+1 ⎩ 0 if |y| > π/ h.
(2.3)
2638
Then
F. Stenger / Linear Algebra and its Applications 429 (2008) 2636–2639
2π(π/2 − ε) |F (x + iy) − Fh (x + iy)| < Cε exp − , h
(2.4)
where Cε is a constant depending only on ε. Proof. By way of explanation of our form of Fh we may note that considered as a function of y, Fh is the Fourier transform, Fh (x + iy) = fh (τ )eiyτ dτ, (2.5) R
where fh (τ ) denotes the Whittaker cardinal series, ∞ sin πh (τ − kh) fh (τ ) = . f (kh) π h (τ − kh)
(2.6)
k=−∞
Here f denotes the function f (τ ) =
exτ , 1 + exp(eτ )
(2.7)
so the series representation of fh converges uniformly and absolutely, for all z = x + iy ∈ Dε . We may also note here, that for y real, ikhy he if |y| < π/ h, iyτ e S(k, h)(τ )dτ = (2.8) 0 if |y| > π/ h. R It is now easy to deduce that Fh does in fact converge to F uniformly on the strip Dε , as h → 0. It follows, therefore, that Fh converges to F at all points of the critical strip, D. Let us next deduce a sequence of Fourier polynomials that converges to F . To this end, we select an arbitrary positive integer N , and proceed as in [2, Section 2.2], to get Theorem 2.2. Set 1/2 h = π(π/2−ε) , xN ⎧ N ⎨ Fh (x, y) = k=−N ⎩ 0
exp(nhx) ikhy e exp(enh )+1
(2.9)
if |y| > π/ h.
Then, for all x + iy ∈ Dε , |F (x + iy) − Fh (x, y)| < Cε N
if |y| < π/ h,
1/2
π(π/2 − ε) 1/2 , exp − xN
(2.10)
where Cε is a constant depending only on ε. For example, by Hurwitz’s theorem, Riemann’s hypothesis could be proved if it could be shown that for every ε ∈ (0, 1/2), the only zeros of Fh (x, y) occurring in the region Bε,h = {z ∈ C : ε < z < 1 − ε, |z| < π/ h − ε} must lie on the line z = 1/2.
F. Stenger / Linear Algebra and its Applications 429 (2008) 2636–2639
Acknowledgments The author is grateful to the referees for valuable criticisms. References [1] Richard S. Varga, Scientific Computations on Mathematical Problems and Conjectures, SIAM, 1990. [2] Frank Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, 1993.
2639
Linear Algebra and its Applications 429 (2008) 2636–2639 www.elsevier.com/locate/laa
Fourier series for zeta function via Sinc Frank Stenger School of Computing, University of Utah, Salt Lake City, UT 84112, USA Received 30 June 2006; accepted 16 January 2008 Available online 11 March 2008 Submitted by A. Frommer This paper is presented in honor of Richard Varga’s 80th birthday
Abstract In this paper we derive some Fourier series and Fourier polynomial approximations to a function F which has the same zeros as the zeta function, ζ (z) on the strip {z ∈ C : 0 < z < 1}. These approximations depend on an arbritrary positive parameter h, and which for arbitrary ε ∈ (0, 1/2), converge uniformly to ζ (z) on the rectangle {z ∈ C : ε < z < 1 − ε, −π/ h < z < π/ h}. © 2008 Elsevier Inc. All rights reserved. Keywords: Riemann zeta function; Zeros; Fourier series; Sinc methods
1. Results This paper is presented in honor of Richard Varga, who has made important contributions to the determination of the moments of the Riemann ζ -function and the de Bruijn–Newman constant [1, Chapter 3]. The usual series expansion of the zeta function can be obtained via term-wise integration of the expression ∞ 1 nz n=1 ∞ z−1 t 1 = dt (z) 0 et − 1
ζ (z) =
E-mail address: [email protected] 0024-3795/$ - see front matter ( 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2008.01.037
(1.1)
F. Stenger / Linear Algebra and its Applications 429 (2008) 2636–2639
=
1 (z)
∞
2637
e−t t z−1 (1 + e−t + e−2t + · · ·)dt,
0
where (z) denotes the usual gamma function. Unfortunately, the above infinite series converges only in the region z > 1, and it is thus not convenient for studying the zeros of the zeta function in the critical strip, D = {z ∈ C : 0 < z < 1}. However, the above series for ζ (z) immediately yields ∞
2 2 ζ (z) = . z 2 (2n)z
(1.2)
n=1
By subtracting the series (1.2) from the series in (1.1), and then proceeding as in (1.1) above, we immediately get ∞ z−1 t 2 dt, (1.3) F (z) ≡ 1 − z (z) ζ (z) = 2 et + 1 0 and the integral representation in this expression converges to F (z) in the region {z ∈ C : z > 0}. Note also, that the factor multiplying ζ (z) in (1.3) does not have any zeros in the critical strip, D, so that both of the functions F (z) and ζ (z) have the same zeros in D. 2. Application of Sinc quadrature The conditions for convergence of Sinc quadrature are quite simple [2, Section 4.2]. Assume that the function g in the integral ∞ I= g(t)dt. (2.1) 0
satisfies the following assumptions: 1. g is analytic in the sector, Sε ≡ {t ∈ C : | arg(t)| < π/2 − ε}, with ε an arbitrary positive constant in the interval (0, 1), and 2. g ≡ ∂Sε |g(t)dt| < ∞. Then, for arbitrary bounded h > 0, we have [2, Theorem 4.2.2 (b)] ∞ 2π(π/2 − ε) kh kh e g e < C exp − , I − h h
(2.2)
k=−∞
where C is a constant that is proportional to g. Now, by taking g(t) = t z−1 /(exp(t) + 1), we get a function g which not only depends on t but also on a complex parameter z. Furthermore, if z ∈ Dε = {z ∈ C : ε < z < 1 − ε}, (where we assume that 0 < ε < 1/2) then this function g does, in fact, satisfy all of the above conditions, enabling us to arrive at the following result. Theorem 2.1. Let F be defined by the integral (1.3), let z = x + iy ∈ Dε , and set ⎧ ∞ ⎨ exp(nhx) eikhy if |y| < π/ h, Fh (x + iy) = k=−∞ exp(enh )+1 ⎩ 0 if |y| > π/ h.
(2.3)
2638
Then
F. Stenger / Linear Algebra and its Applications 429 (2008) 2636–2639
2π(π/2 − ε) |F (x + iy) − Fh (x + iy)| < Cε exp − , h
(2.4)
where Cε is a constant depending only on ε. Proof. By way of explanation of our form of Fh we may note that considered as a function of y, Fh is the Fourier transform, Fh (x + iy) = fh (τ )eiyτ dτ, (2.5) R
where fh (τ ) denotes the Whittaker cardinal series, ∞ sin πh (τ − kh) fh (τ ) = . f (kh) π h (τ − kh)
(2.6)
k=−∞
Here f denotes the function f (τ ) =
exτ , 1 + exp(eτ )
(2.7)
so the series representation of fh converges uniformly and absolutely, for all z = x + iy ∈ Dε . We may also note here, that for y real, ikhy he if |y| < π/ h, iyτ e S(k, h)(τ )dτ = (2.8) 0 if |y| > π/ h. R It is now easy to deduce that Fh does in fact converge to F uniformly on the strip Dε , as h → 0. It follows, therefore, that Fh converges to F at all points of the critical strip, D. Let us next deduce a sequence of Fourier polynomials that converges to F . To this end, we select an arbitrary positive integer N , and proceed as in [2, Section 2.2], to get Theorem 2.2. Set 1/2 h = π(π/2−ε) , xN ⎧ N ⎨ Fh (x, y) = k=−N ⎩ 0
exp(nhx) ikhy e exp(enh )+1
(2.9)
if |y| > π/ h.
Then, for all x + iy ∈ Dε , |F (x + iy) − Fh (x, y)| < Cε N
if |y| < π/ h,
1/2
π(π/2 − ε) 1/2 , exp − xN
(2.10)
where Cε is a constant depending only on ε. For example, by Hurwitz’s theorem, Riemann’s hypothesis could be proved if it could be shown that for every ε ∈ (0, 1/2), the only zeros of Fh (x, y) occurring in the region Bε,h = {z ∈ C : ε < z < 1 − ε, |z| < π/ h − ε} must lie on the line z = 1/2.
F. Stenger / Linear Algebra and its Applications 429 (2008) 2636–2639
Acknowledgments The author is grateful to the referees for valuable criticisms. References [1] Richard S. Varga, Scientific Computations on Mathematical Problems and Conjectures, SIAM, 1990. [2] Frank Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, 1993.
2639