Complex integration, method of

method of contour integration

One of the universal methods in the study and applications of zeta-functions, $ L $- functions (cf. Zeta-function; $ L $- function) and, more generally, functions defined by Dirichlet series.

The method of complex integration was first introduced by B. Riemann [1] in 1876 into number theory in connection with the study of the properties of the zeta-function. The well-known modern applications of the method of complex integration, which use the Cauchy theorem on residues, the Phragmén–Lindelöf theorem on Dirichlet series, the saddle point method, etc., are extremely varied in their form and content. The method of complex integration is used for the analytic continuation and the derivation of functional equations of Dirichlet functions; for the derivation of approximate functional equations of these functions; for estimating the number of their non-trivial zeros and the density of the distribution of these zeros in some part of the critical strip; and for obtaining asymptotic formulas and estimates of various kinds of the most important arithmetic functions. A classical example of the method of complex integration is illustrated by the proof of the analytic continuation and the derivation of the functional equation of the Riemann zeta-function (see [2], [3]). For $ s = \sigma + i t $, $ \sigma > 0 $, one has

$$ n ^ {-s} \Gamma ( s) = \ n ^ {-s} \int\limits _ { 0 } ^ \infty e ^ {-x} d x = \int\limits _ { 0 } ^ \infty e ^ {-nx} x ^ {s-1} d x . $$

after summation one finds that the function $ \zeta ( s) $, originally defined by the series $ \zeta ( s) = \sum _ {n=1} ^ \infty n ^ {-s} $ for $ \sigma > 1 $, is also expressed by the formula

$$ \tag{1 } \zeta ( s) = \ \frac{1}{\Gamma ( s) } \int\limits _ { 0 } ^ \infty \frac{x ^ {s-1} }{e ^ {x} - 1 } d x . $$

Consider the integral

$$ J ( s) = \ \frac{1}{2 \pi i } \int\limits _ { C } \frac{z ^ {s-1} }{e ^ {-z} - 1 } d z , $$

taken along the (infinite) contour $ C = \alpha + \beta + \gamma $, where $ \alpha , \gamma $ pass along the lower and upper edges of the cut along the negative real axis of the $ z $- plane and $ \beta $ is a circle of radius $ r < 2 \pi $ passing round the origin. The integral $ J ( s) $ converges for all $ s $, the convergence being uniform in any disc $ | s | < \Delta $, since the integrand is less than $ e ^ {- 0 .5 | z | } $ on $ \alpha $ and $ \gamma $ for all $ | x | > z _ {0} ( \Delta ) $. By Cauchy's theorem it does not depend on $ r $, hence is an entire function of $ s $. Assuming that on $ \alpha , \beta $ and $ \gamma $, respectively, $ z = \delta e ^ {- i \pi } $, $ z = r e ^ {i \theta } $, $ z = \delta e ^ {i \pi } $, and $ f ( z) = 1 / ( e ^ {z} - 1 ) $, it is easy to write $ J ( s) $ as a sum of integrals with respect to real variables:

$$ \pi J ( s) = \sin \ \pi s \int\limits _ { r } ^ \infty \delta ^ {s-1} f ( - \delta ) d \delta + \frac{r ^ {s} }{2} \int\limits _ {- \pi } ^ \pi e ^ {i \theta s } f ( r e ^ {i \theta } ) d \theta . $$

In the disc $ | z | < \pi $ one has $ | z f ( z) | < A $. Therefore the second term on the right-hand side of the equation is less than $ 2 \pi Ar ^ {\sigma - 1 } e ^ {\pi | t | } $, which for any fixed $ s $ with $ \sigma > 1 $ converges to zero as $ r \rightarrow 0 $. Hence, by formula (1), $ \pi J ( s) = \Gamma ( s) \zeta ( s) \sin \pi s $ and

$$ \tag{2 } \zeta ( s) = \ \frac{\pi J ( s) }{\Gamma ( s) \sin \pi s } = \Gamma ( 1 - s ) J ( s) . $$

This formula, proved under the hypothesis that $ \sigma > 1 $, provides a continuation of $ \zeta ( s) $ to the whole plane. It is clear from this that $ \zeta ( s) $ is a single-valued analytic function on the whole $ s $- plane, having as its only singularity a simple pole at the point $ s = 1 $ with residue $ 1 $.

To deduce the functional equation of $ \zeta ( s) $ it is assumed that $ \sigma < 0 $ and that $ N $ is an integer $ > 4 $. Let

$$ J _ {N} ( s) = \ \frac{1}{2 \pi i } \int\limits _ {C ( N) } \frac{z ^ {s-1} }{e ^ {-z} - 1 } d z , $$

where $ C ( N) $ is a contour that differs from the previous contour by joining $ \alpha $ and $ \gamma $ by a circular arc of radius $ R = 2 N + 1 $ with centre at the origin. The integral along the outer arc of the contour $ C ( N) $ can be estimated in the form $ A R ^ \sigma e ^ {\pi | t | } $, which for $ \sigma < 0 $ converges to zero as $ N \rightarrow \infty $. Hence $ J _ {N} ( s) \rightarrow J ( s) $ as $ N \rightarrow \infty $. On the other hand, by the residue theorem,

$$ J _ {N} ( s) = \sum_{n=1} ^ { N } \{ ( 2 \pi n i ) ^ {s-1} + ( - 2 \pi n i ) ^ {s-1} \} = $$

$$ = \ 2 ( 2 \pi ) ^ {s-1} \sin \frac{\pi s }{2} \sum_{n=1} ^ { N } n ^ {s-1}. $$

Therefore, when $ \sigma < 0 $,

$$ J ( s) = \lim\limits \ J _ {N} ( s) = 2 ( 2 \pi ) ^ {s-1} \zeta ( 1 - s ) \ \sin \ \frac{\pi s }{2} . $$

This equation, in combination with formula (2), gives the relation

$$ \zeta ( 1 - s ) = 2 ( 2 \pi ) ^ {-s} \Gamma ( s) \zeta ( s) \cos \ \frac{\pi s }{2} , $$

which, according to the theory of analytic continuation, holds throughout the whole $ s $- plane. It is called the functional equation of the Riemann zeta-function.

The method of complex integration plays an important role in obtaining the approximate functional equations which lie at the basis of modern estimates of Dirichlet functions (see [4], [5]).

The method of complex integration is fundamental in the study of the distribution of zeros of the functions $ \zeta ( s) $, $ L ( s , \chi ) $, etc. Until recently it has been applied in the form of the well-known Littlewood theorem on the number of zeros in a rectangle of a function $ F ( s) $ that is regular for $ \sigma > 0 $, the Bäcklund theorem about $ \mathop{\rm arg} F ( s) $, and also to theorems about convexity of mean values of analytic functions (see [2]). In 1969, H. Montgomery [6] found a new, direct, more powerful way of applying the method of complex integration to these ends.

The method of complex integration in its applications to number theory naturally arises in connection with summation formulas for the coefficients of Dirichlet series (see [2], [7]).

References

[1]	H. Weber (ed.) , Riemann's gesammelte math. Werke , Teubner (1892) (Dover, reprint, 1953)
[2]	E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)
[3]	K. Prachar, "Primzahlverteilung" , Springer (1957)
[4]	A.F. Lavrik, "Functional equations of Dirichlet functions" Izv. Akad. Nauk SSSR Ser. Mat. , 31 : 2 (1967) pp. 431–442 (In Russian)
[5]	A.F. Lavrik, "Approximate functional equations of Dirichlet functions" Izv. Akad. Nauk SSSR Ser. Mat. , 32 : 1 (1968) pp. 134–185 (In Russian)
[6]	H. Davenport, "Multiplicative number theory" , Springer (1980)
[7]	A.A. Karatsuba, "A uniform estimate for the remainder term in Dirichlet's divisor problem" Izv. Akad. Nauk SSSR Ser. Mat. , 36 : 3 (1972) pp. 475–483 (In Russian)

Comments

The derivation of the functional equation for $ \zeta ( s) $ can be found in [2] (especially Section 2.4), in which also a number of other derivations of the functional equation are given.

The article above gives one nice illustration of the "method of contour integration" . The philosophy behind this method is as follows. Suppose one has to evaluate an integral over a (smooth or rectifiable) contour. (A contour is a curve given by a parametrization $ z ( t) = x ( t) + i y ( t) $, $ t \in [ a , b ] \subset \mathbf R $, such that $ x ( t) $ and $ y ( t) $ are continuously differentiable on each interval $ [ x _ {k} , x _ {k+1} ] $ of a finite partition $ a = x _ {1} < x _ {2} < \dots < x _ {n} = b $ of $ [ a , b ] $. A contour is called smooth if $ x ^ \prime ( t) $ and $ y ^ \prime ( t) $ do not simultaneously vanish, except at a finite number of points, see also Rectifiable curve.) The method of contour integration is to shift this contour (or extend it, like in the article above) in such a way that the integral can be easily evaluated (in most cases by the residue theorem, cf. Cauchy integral), and then to estimate the difference between the integrals over the shifted contour and the original contour (if the integrand is analytic one can sometimes use the Cauchy integral theorem; if the contour has been extended, subtle estimation of the integrand might give a result). Applications of this method to certain types of integrals (e.g., of the form $ \int _ {- \infty } ^ \infty ( P ( x) / Q ( x) ) dx $, $ \int _ {0} ^ \infty ( P ( x) / Q ( x) ) d x $, $ \int _ {- \infty } ^ \infty ( P ( x) / Q ( x)) \sin x d x $, etc.) and in the proof of many important theorems in complex analysis can be found in almost any textbook on complex analysis, e.g. [a2], [a3], [a4]. Reference [a5] contains a wealth of formulas derived by the method of contour integration.

A novel approach to the prime number theorem using the method of contour integration was given in [a1].

The method of contour integration is also called the contour integral method.

References

[a1]	D.J. Newman, "Simple analytic proof of the prime number theorem" Amer. Math. Monthly , 11 (1980) pp. 693–696
[a2]	J. Bak, D.J. Newman, "Complex analysis" , Springer (1982)
[a3]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241
[a4]	M.A. Evgrafov, "Analytic functions" , Dover, reprint , Philadelphia (1978) (Translated from Russian)
[a5]	D.S. Mitrinović, J.D. Kečkić, "The Cauchy method of residues: theory and applications" , Reidel (1984)