1. 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula Euler-Maclaurin integration formula : B2B2 B4B4 B6B6 B8B8 1/6  1/30 1/42  1/30 Let   

2.   

3.    Stirling’s series

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

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

6. Definition : Contour Integral  0 for Re z >1 diverges for Re z <1  agrees with integral representation for Re z > 1 C1C1

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

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

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

10. Riemann’s functional equation :  fortrivial zeros converges for Re z > 1   (z) is regular for Re z < 0.  (0) diverges   (1) diverges while  (0) is indeterminate. Since the integrand inis 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 )  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 line Re 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 1.Incomplete Gamma Functions 2.Incomplete Beta Functions 3.Exponential Integral 4.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) For non-integral a : See Ex 1.3.3 & Ex.13.6.4 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 E 1, Ei analytic continued. Branch-cut : (  x)–axis. Mathematica

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

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

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

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