***** Quadruple precision version: precision forced to 128 bits main steps: ALL runComputation: steps: ALL ****** GETTING DIVISORS FROM FILE Plan generation 0.000000 min. 1.000000 sec. 292.610000 millisec. COMPUTE ODD LNGammaOdd computation 0.000000 min. 4.000000 sec. 691.174000 millisec. -temp-LGodd.qd 0.000000 min. 0.000000 sec. 24.792000 millisec. FFT(LGodd) 0.000000 min. 11.000000 sec. 259.579000 millisec. -temp-fftLGodd.qd 0.000000 min. 0.000000 sec. 25.858000 millisec. AK computation 0.000000 min. 1.000000 sec. 369.455000 millisec. -temp-ak.qd 0.000000 min. 0.000000 sec. 25.594000 millisec. FFT(AK) 0.000000 min. 11.000000 sec. 242.621000 millisec. -temp-fftAK.qd 0.000000 min. 0.000000 sec. 26.141000 millisec. Computing the sum and minmax over odd characters *** indexes for minimal and maximal odd characters L'/L (1,chi) chiminodd index = 40677 chimaxodd index = 350813 Kummer_realcorrection = -2.1538725705742171360594603007e+07 Kummer_res = +2.1538725218437500483175180637e+07 Kummer (r) = +6.1427985125041165184921470949e-01 log r = -4.8730467087741942237032095767e-01 H_correction = -1.5875956599358387881627771685e+07 log h1 = +5.6627686190791126015474089514e+06 log Pi_odd = -4.8730467087741942237032095767e-01 *** indexes for minimal and maximal odd characters L (1,chi) chiminoddL index = 1727423 chimaxoddL index = 1784553 *** indexes for minimal and maximal odd characters L'(1,chi) chiminoddLprime index = 40677 chimaxoddLprime index = 298563 *** indexes for minimal and maximal odd characters |b_chi| chiminoddbchi index = 298563 chimaxoddbchi index = 208845 *** First: oddresult = -2515451.247998100234742889758138647517 , (res = -1.207541990199350413223496211753 ) OddResult computation 0.000000 min. 1.000000 sec. 974.418000 millisec. COMPUTE EVEN loadTable(): initialize table INFO: precomputed table allocated 138 elements INFO: init() time (seconds) = 0.000007 S computation 0.000000 min. 7.000000 sec. 168.662000 millisec. -temp-S.qd 0.000000 min. 0.000000 sec. 25.002000 millisec. Sfft 0.000000 min. 11.000000 sec. 181.508000 millisec. -temp-fftS.qd 0.000000 min. 0.000000 sec. 24.092000 millisec. mkLGeven 0.000000 min. 1.000000 sec. 624.827000 millisec. -temp-LGeven.qd 0.000000 min. 0.000000 sec. 22.794000 millisec. LGevenfft 0.000000 min. 11.000000 sec. 131.536000 millisec. -temp-fftLGeven.qd 0.000000 min. 0.000000 sec. 23.828000 millisec. Computing the sum and minmax over even characters ... *** indexes for minimal and maximal even characters L'/L (1,chi) chimineven index = 75742 chimaxeven index = 728946 *** indexes for minimal and maximal even characters L(1,chi) chiminevenL index = 1857478 chimaxevenL index = 126690 *** indexes for minimal and maximal even characters L'(1,chi) chiminevenLprime index = 75742 chimaxevenLprime index = 126690 *** indexes for minimal and maximal even characters |b_chi| chiminevenbchi index = 126690 chimaxevenbchi index = 225638 log Pi_even = -3.0315015018023482364565843007e+00 *** Second: evenresult = -2515446.250770336601282208522190628315 , (res = -5030892.501540673202564417044381256630 ) EvenResult computation 0.000000 min. 1.000000 sec. 880.186000 millisec. Inverse plan generation time 0.000000 min. 1.000000 sec. 298.683000 millisec. INVERSE LGodd FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 66.765000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 11.000000 sec. 259.061000 millisec. -temp-fftLGodd_INV.qd 0.000000 min. 0.000000 sec. 24.358000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 24.418000 millisec. INVERSE AK FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 46.709000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 11.000000 sec. 220.400000 millisec. -temp-fftAK_INV.qd 0.000000 min. 0.000000 sec. 25.961000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 26.020000 millisec. INVERSE S FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 74.992000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 11.000000 sec. 245.643000 millisec. -temp-fftS_INV.qd 0.000000 min. 0.000000 sec. 24.725000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 24.781000 millisec. INVERSE LGeven FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 104.385000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 11.000000 sec. 204.419000 millisec. -temp-fftLGeven_INV.qd 0.000000 min. 0.000000 sec. 24.678000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 24.736000 millisec. NORM OF S ... Measures for S NDIFF_2 = Norm2(S - fft(fft(S)) = +5.2833497398116012335801982475e-30 NBASE_2 = Norm2(S) = +7.0921034410368515562328330409e+03 Inverse precision for S in Norm2: NDIFF_2 / NBASE_2 = +7.4496230684407303376755596697e-34 NDIFF_1 = Norm1(S - fft(fft(S)) = +4.7606682437726472081216600130e-27 NBASE_1 = Norm1(S) = +4.1793406528728330432894800322e+06 Inverse precision for S in Norm1: NDIFF_1 / NBASE_1 = +1.1390955270660255795380392061e-33 NDIFF_0 = Norm00(S - fft(fft(S)) = +3.7403627684949773316962196583e-32 NBASE_0 = Norm00(S) = +2.1168433872021094702661013830e+02 Inverse precision for S in Norm0: NDIFF_0 / NBASE_0 = +1.7669529976134504641856460984e-34 NORM OF AK ... Measures for AK NDIFF_2 = Norm2(AK - fft(fft(AK)) = +4.5207446797699695156649120811e-31 NBASE_2 = Norm2(AK) = +5.8922463175500961339882664274e+02 Inverse precision for AK in Norm2: NDIFF_2 / NBASE_2 = +7.6723620095529618838634151462e-34 NDIFF_1 = Norm1(AK - fft(fft(AK)) = +4.0874891519284317156016667822e-28 NBASE_1 = Norm1(AK) = +5.2077875000012001246209406385e+05 Inverse precision for AK in Norm1: NDIFF_1 / NBASE_1 = +7.8488017261216778940807910872e-34 NDIFF_0 = Norm00(AK - fft(fft(AK)) = +1.6567475823156319987138090584e-33 NBASE_0 = Norm00(AK) = +9.9999903990030324748921928053e-01 Inverse precision for AK in Norm0: NDIFF_0 / NBASE_0 = +1.6567491729600105525929743602e-33 NORM OF LGODD ... Measures for LGodd NDIFF_2 = Norm2(LGodd - fft(fft(LGodd)) = +1.4165637264602930054719974036e-30 NBASE_2 = Norm2(LGodd) = +1.8440616818512959652036241781e+03 Inverse precision for LGodd in Norm2: NDIFF_2 / NBASE_2 = +7.6817589151257244114962310368e-34 NDIFF_1 = Norm1(LGodd - fft(fft(LGodd)) = +1.2799832397853614113632024906e-27 NBASE_1 = Norm1(LGodd) = +1.4342202899019962347790436034e+06 Inverse precision for LGodd in Norm1: NDIFF_1 / NBASE_1 = +8.9245930265902582045485922101e-34 NDIFF_0 = Norm00(LGodd - fft(fft(LGodd)) = +5.3102956377777236786845831127e-33 NBASE_0 = Norm00(LGodd) = +1.4549375333468143654393508152e+01 Inverse precision for LGodd in Norm0: NDIFF_0 / NBASE_0 = +3.6498444201672163489041935793e-34 NORM OF LGEVEN ... Measures for LGeven NDIFF_2 = Norm2(LGeven - fft(fft(LGeven)) = +8.6131712629889524973355417843e-31 NBASE_2 = Norm2(LGeven) = +1.1648823826527037313592573973e+03 Inverse precision for LGeven in Norm2: NDIFF_2 / NBASE_2 = +7.3940265483067832847404965461e-34 NDIFF_1 = Norm1(LGeven - fft(fft(LGeven)) = +7.7766669447085100015309432969e-28 NBASE_1 = Norm1(LGeven) = +7.2194571640171117142182198864e+05 Inverse precision for LGeven in Norm1: NDIFF_1 / NBASE_1 = +1.0771816728089499301563738050e-33 NDIFF_0 = Norm00(LGeven - fft(fft(LGeven)) = +3.6026284748159908830758231401e-33 NBASE_0 = Norm00(LGeven) = +1.3404646001803707423611521912e+01 Inverse precision for LGeven in Norm0: NDIFF_0 / NBASE_0 = +2.6875968782250773642085369439e-34 *** RESULTS: -------------- gamma_Kr constants for every intermediate field ---- gammaK_1(2083117) = 18.973431888336510823388104720474 gammaKplus_1(2083117) = 11.066391292780313030022555796085 ---- gammaK_2(2083117) = 11.066391292780313030022555796085 gammaKplus_2(2083117) = 13.055126789107473850110026352197 ---- gammaK_3(2083117) = 7.207289679421828705297083219309 gammaKplus_3(2083117) = 9.809253741236625869673419822229 ---- gammaK_6(2083117) = 9.809253741236625869673419822229 gammaKplus_6(2083117) = 11.990097823828845798445269330984 ---- gammaK_7(2083117) = 15.013259562338554462713574031316 gammaKplus_7(2083117) = 14.731661705499587486477740835806 ---- gammaK_14(2083117) = 14.731661705499587486477740835806 gammaKplus_14(2083117) = 8.454884884757930867463957465660 ---- gammaK_21(2083117) = 10.763750622478079315461477900984 gammaKplus_21(2083117) = 12.611668885926717536982322239647 ---- gammaK_42(2083117) = 12.611668885926717536982322239647 gammaKplus_42(2083117) = 10.783869442620073336072113173465 ---- gammaK_24799(2083117) = 3.904205114343354820829652696328 gammaKplus_24799(2083117) = 2.601335485700214438431125029889 ---- gammaK_49598(2083117) = 2.601335485700214438431125029889 gammaKplus_49598(2083117) = 2.564697851272338841059461276159 ---- gammaK_74397(2083117) = 1.845562487508117443709252097529 gammaKplus_74397(2083117) = -0.206981263105789172392650935278 ---- gammaK_148794(2083117) = -0.206981263105789172392650935278 gammaKplus_148794(2083117) = -3.286606594769830568715971732234 ---- gammaK_173593(2083117) = 0.167333838515374688535334477108 gammaKplus_173593(2083117) = -0.909038860990594178546892864062 ---- gammaK_347186(2083117) = -0.909038860990594178546892864062 gammaKplus_347186(2083117) = -0.176391492255247444557321895833 ---- gammaK_520779(2083117) = 2.389328937490657671835190371157 gammaKplus_520779(2083117) = 0.545416356937839235493187153604 ---- gammaK_1041558(2083117) = 0.545416356937839235493187153604 gammaKplus_1041558(2083117) = 0.577215664901532865549427242513 -------------- Other quantities -------------- [1] EK(2083117) = 18.973431888336510823388104720474 [2] EKplus(2083117) = 11.066391292780313030022555796085 [3] EK(2083117)diff = 7.907040595556197793365548924390 -------------- [4] min_even(2083117) = 0.002069931040548859200592895418 [5] min_odd(2083117) = 0.000564732617871719275167863170 [6] min_{chi neq chi_0}|L'/L(1,chi)|(2083117) = 0.000564732617871719275167863170 -------------- [7] max_even(2083117) = 2.646077317545001681519120545783 [8] max_odd(2083117) = 2.777151127509787894670486152518 [9] max_{chi neq chi_0}|L'/L(1,chi)|(2083117) = 2.777151127509787894670486152518 -------------- [10] Kummer_ratio(2083117) = 0.614279851250411651849214709491 [11] log(Kummer) = log r(2083117) = -0.487304670877419422370320957667 [12] logh_1(2083117) = 5662768.619079112601547408951405929037 -------------- [13] Pi(2083117,1.0Q) = 33.744119755756867593040160497117 [14] Laurent series of zeta_{Q(zeta_q)} at 1: c_{-1} = Pi(q,1)^(-1) = 0.029634792883562962735815896025 [15] c_{0} = c_{-1} * EK = 0.562273724301241416781668760118 -------------- [16] minL_even(2083117) = 0.254645298051542885339855621530 [17] minL_odd(2083117) = 0.265410656083829754629503215934 [18] minL_{chi neq chi_0}|L(1,chi)|(2083117) = 0.254645298051542885339855621530 -------------- [19] maxL_even(2083117) = 5.378033559016930896100312614754 [20] maxL_odd(2083117) = 5.502242677380206866570934906557 [21] maxL_{chi neq chi_0}|L(1,chi)|(2083117) = 5.502242677380206866570934906557 -------------- [22] minLprime_even(2083117) = 0.001233495526045247437255611096 [23] minLprime_odd(2083117) = 0.000697080087028676368577656773 [24] minLprime_{chi neq chi_0}|Lprime(1,chi)|(2083117) = 0.000697080087028676368577656773 -------------- [25] maxLprime_even(2083117) = 12.889283055522601195851089607551 [26] maxLprime_odd(2083117) = 14.509919472956543522418966013161 [27] maxLprime_{chi neq chi_0}|Lprime(1,chi)|(2083117) = 14.509919472956543522418966013161 -------------- [28] minbchi_even(2083117) = 9.738723711755752103288106377941 [29] minbchi_odd(2083117) = 9.514143388930533911636919569290 [30] minbchi_{chi neq chi_0}(2083117) = 9.514143388930533911636919569290 -------------- [31] maxbchi_even(2083117) = 14.533296565201230271623080764217 [32] maxbchi_odd(2083117) = 14.556110793732919098858674169074 [33] maxbchi_{chi neq chi_0}(2083117) = 14.556110793732919098858674169074 -------------- ******** the quadratic character is even [34] L(1,chi_quad)(2083117) = 0.902843171622927894131321609182 [35] L(1,chi_quad)/log(2083117) = 0.062053738840380243112367950354 [36] L(1,chi_quad)/loglog(2083117) = 0.337190272014090856721455886916 [37] Lprime(1,chi_quad)(2083117) = -0.028709788057355385777133747933 [38] Lderlog(1,chi_quad)(2083117) = -0.031799307963693630056240088909 -------------- *** A priori (closed formulas) norms *** N1LGeven = +7.2194571640171169022174983259e+05 N00LGeven = +1.3404646001803707423611521912e+01 N00LGodd = +1.4549375333468143607900670670e+01 N1Seven = +4.1793406528728331774353487195e+06 N00Seven = +2.1168433872021094702661013830e+02 N1AKodd = +5.2077875000012001246209406385e+05 N00AKodd = +9.9999903990030324748921928053e-01 N2AKodd = +5.8922463175500961339882664274e+02 TOTAL TIME 1.000000 min. 58.000000 sec. 30.626000 millisec.