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|krj| | ks| |kr d t  t||d }| | krˆ d t  t|| d }t|jd }j|8_| |fS)NrrrMcsB|dd|d||d|S)NrrMrr)rr}rrrZrrrr;vs z*secondzeta_singular_term..rrrrrr?)r}rrrr"r~r.r$r\r!r1r[rr%r)rr'rrcfactorrrrrZmg1rrZmg2Zpolestrrrrsecondzeta_singular_termpsB. . :   " "rQ?cKs||}||}|j}||r||dkrt|d|dkrL|jStt||}|d@rl|jSd| d|d|j | ddd| dS|j }zt |||}t | |d }|j |d7_ t|||d d d \} } } t|||d d d \} } }t|||d d \}}t |||}| | |}| | ||}|d rtd| td| dtd| td|dtd|td|W5||_ X|drt | t|d }t |d||jdd|}| |fS| S)a Evaluates the secondary zeta function `Z(s)`, defined for `\mathrm{Re}(s)>1` by .. math :: Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s} where `\frac12+i\tau_n` runs through the zeros of `\zeta(s)` with imaginary part positive. `Z(s)` extends to a meromorphic function on `\mathbb{C}` with a double pole at `s=1` and simple poles at the points `-2n` for `n=0`, 1, 2, ... **Examples** >>> from mpmath import * >>> mp.pretty = True; mp.dps = 15 >>> secondzeta(2) 0.023104993115419 >>> xi = lambda s: 0.5*s*(s-1)*pi**(-0.5*s)*gamma(0.5*s)*zeta(s) >>> Xi = lambda t: xi(0.5+t*j) >>> chop(-0.5*diff(Xi,0,n=2)/Xi(0)) 0.023104993115419 We may ask for an approximate error value:: >>> secondzeta(0.5+100j, error=True) ((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16) The function has poles at the negative odd integers, and dyadic rational values at the negative even integers:: >>> mp.dps = 30 >>> secondzeta(-8) -0.67236328125 >>> secondzeta(-7) +inf **Implementation notes** The function is computed as sum of four terms `Z(s)=A(s)-P(s)+E(s)-S(s)` respectively main, prime, exponential and singular terms. The main term `A(s)` is computed from the zeros of zeta. The prime term depends on the von Mangoldt function. The singular term is responsible for the poles of the function. The four terms depends on a small parameter `a`. We may change the value of `a`. Theoretically this has no effect on the sum of the four terms, but in practice may be important. A smaller value of the parameter `a` makes `A(s)` depend on a smaller number of zeros of zeta, but `P(s)` uses more values of von Mangoldt function. We may also add a verbose option to obtain data about the values of the four terms. >>> mp.dps = 10 >>> secondzeta(0.5 + 40j, error=True, verbose=True) main term = (-30190318549.138656312556 - 13964804384.624622876523j) computed using 19 zeros of zeta prime term = (132717176.89212754625045 + 188980555.17563978290601j) computed using 9 values of the von Mangoldt function exponential term = (542447428666.07179812536 + 362434922978.80192435203j) singular term = (512124392939.98154322355 + 348281138038.65531023921j) ((0.059471043 + 0.3463514534j), 1.455191523e-11) >>> secondzeta(0.5 + 40j, a=0.04, error=True, verbose=True) main term = (-151962888.19606243907725 - 217930683.90210294051982j) computed using 9 zeros of zeta prime term = (2476659342.3038722372461 + 28711581821.921627163136j) computed using 37 values of the von Mangoldt function exponential term = (178506047114.7838188264 + 819674143244.45677330576j) singular term = (175877424884.22441310708 + 790744630738.28669174871j) ((0.059471043 + 0.3463514534j), 1.455191523e-11) Notice the great cancellation between the four terms. Changing `a`, the four terms are very different numbers but the cancellation gives the good value of Z(s). **References** A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier, 53, (2003) 665--699. A. Voros, Zeta functions over Zeros of Zeta Functions, Lecture Notes of the Unione Matematica Italiana, Springer, 2009. rr{rrr7TrrrTrue)rr)rrz main term =z computed usingz zeros of zetaz prime term =z#values of the von Mangoldt functionzexponential term =zsingular term =rr)rr~r&rr\r$r%roZfractionrr"rrrrrrr[r)rr'rrcrrr"Zt3rrdr1gtrer2ptZt4Zr4rr2rrrr secondzetasH\   $            rc s|dkr S|dkr&Sdkr>||Sdkrƈr^tdtd}j}j}t |D] }|||7}||9}q| |||S |dd d  d|}ddj fdd}|d|djg7}|sss|dkr|}|S)a Gives the Lerch transcendent, defined for `|z| < 1` and `\Re{a} > 0` by .. math :: \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(a+k)^s} and generally by the recurrence `\Phi(z,s,a) = z \Phi(z,s,a+1) + a^{-s}` along with the integral representation valid for `\Re{a} > 0` .. math :: \Phi(z,s,a) = \frac{1}{2 a^s} + \int_0^{\infty} \frac{z^t}{(a+t)^s} dt - 2 \int_0^{\infty} \frac{\sin(t \log z - s \operatorname{arctan}(t/a)}{(a^2 + t^2)^{s/2} (e^{2 \pi t}-1)} dt. The Lerch transcendent generalizes the Hurwitz zeta function :func:`zeta` (`z = 1`) and the polylogarithm :func:`polylog` (`a = 1`). **Examples** Several evaluations in terms of simpler functions:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> lerchphi(-1,2,0.5); 4*catalan 3.663862376708876060218414 3.663862376708876060218414 >>> diff(lerchphi, (-1,-2,1), (0,1,0)); 7*zeta(3)/(4*pi**2) 0.2131391994087528954617607 0.2131391994087528954617607 >>> lerchphi(-4,1,1); log(5)/4 0.4023594781085250936501898 0.4023594781085250936501898 >>> lerchphi(-3+2j,1,0.5); 2*atanh(sqrt(-3+2j))/sqrt(-3+2j) (1.142423447120257137774002 + 0.2118232380980201350495795j) (1.142423447120257137774002 + 0.2118232380980201350495795j) Evaluation works for complex arguments and `|z| \ge 1`:: >>> lerchphi(1+2j, 3-j, 4+2j) (0.002025009957009908600539469 + 0.003327897536813558807438089j) >>> lerchphi(-2,2,-2.5) -12.28676272353094275265944 >>> lerchphi(10,10,10) (-4.462130727102185701817349e-11 - 1.575172198981096218823481e-12j) >>> lerchphi(10,10,-10.5) (112658784011940.5605789002 - 498113185.5756221777743631j) Some degenerate cases:: >>> lerchphi(0,1,2) 0.5 >>> lerchphi(0,1,-2) -0.5 Reduction to simpler functions:: >>> lerchphi(1, 4.25+1j, 1) (1.044674457556746668033975 - 0.04674508654012658932271226j) >>> zeta(4.25+1j) (1.044674457556746668033975 - 0.04674508654012658932271226j) >>> lerchphi(1 - 0.5**10, 4.25+1j, 1) (1.044629338021507546737197 - 0.04667768813963388181708101j) >>> lerchphi(3, 4, 1) (1.249503297023366545192592 - 0.2314252413375664776474462j) >>> polylog(4, 3) / 3 (1.249503297023366545192592 - 0.2314252413375664776474462j) >>> lerchphi(3, 4, 1 - 0.5**10) (1.253978063946663945672674 - 0.2316736622836535468765376j) **References** 1. 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