13.4. Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

1. 13.4. Sterling’s SeriesDerivation from Euler-Maclaurin Integration Formula Euler-Maclaurin integration formula : Let   

2.   

3.  Stirling’s series  

4. Stirling approx z >> 1 :  A = Arfken’s two-term approx. using Mathematica

5. 13.5. Riemann Zeta Function Riemann Zeta Function : Mathematica Integral representation : Proof :

6. Definition : Contour Integral C1 • 0 for Re z >1 • diverges for Re z <1 agrees with integral representation for Re z > 1 

7. CC1encloses no pole. CC1 encloses all poles. Analytic Continuation Re z > 1 Poles at Similar to ,  Definition valid for all z (except for z integers). Mathematica means n  0  

8. Riemann’s Functional Equation  Riemann’s functional equation

9. Zeta-Function Reflection Formula     zeta-function reflection formula

10. Riemann’s functional equation : converges for Re z > 1  (z) is regular for Re z < 0. (0) diverges  (1) diverges while (0) is indeterminate.  for trivial zeros Since the integrand in is always positive,   (except for the trivial zeros) or i.e., non-trivial zeros of (z) must lie in the critical strip

11. Critical Strip Consider the Dirichlet series : Leibniz criterion  series converges if , i.e.,  for   

12. (0) Simple poles :  

13. Euler Prime Product Formula ( no terms ) ( no terms )  Euler prime product formula

14. Riemann Hypothesis Riemann found a formula that gives the number of primes less than a given number in terms of the non-trivial zeros of (z). Riemann hypothesis : All nontrivial zeros of (z) are on the critical lineRe z ½. • Millennium Prize problems proposed by the Clay Mathematics Institute. • 1. P versus NP • 2. The Hodge conjecture • 3. The Poincaré conjecture (proved by G.Perelman in 2003) • 4. The Riemann hypothesis • 5. Yang–Mills existence and mass gap • 6. Navier–Stokes existence and smoothness • 7. The Birch and Swinnerton-Dyer conjecture

15. 13.6. Other Related Functions Incomplete Gamma Functions Incomplete Beta Functions Exponential Integral Error Function

16. Incomplete Gamma Functions Integral representation:  Exponential integral

17. Series Representation for  (n, x) 

18. Series Representation for (n, x) 

19. Series Representation for  (a, x) &  (a, x) See Ex 1.3.3 & Ex.13.6.4 For non-integral a: Pochhammer symbol Relation to hypergeometric functions: see § 18.6 .

20. Incomplete Beta Functions Ex.13.6.5 Relation to hypergeometric functions: see § 18.5.

21. Exponential Integral Ei(x) P = Cauchy principal value E1 , Ei analytic continued. Branch-cut : (x)–axis. Mathematica

22. Series Expansion For x << 1 : For x >> 1 :

23. Sine & Cosine Integrals not defined Mathematica Ci(z) & li(z) are multi-valued. Branch-cut : (x)–axis. is an entire function

24.   Series expansions : Ex.13.6.13. Asymptotic expansions : § 12.6.

25. Error Function Mathematica Power expansion : Asymptotic expansion (see Ex.12.6.3) :