languasc@zygalski Lquad_analysis % python3.11 analysis-Lquad.py -------------------------------------------------------------------------------- Author of the Script: Alessandro Languasco; (C) 2023 -------------------------------------------------------------------------------- Versions: python: 3.11.3 (main, Apr 7 2023, 19:25:52) [Clang 14.0.0 (clang-1400.0.29.202)] matplotlib: 3.7.1 pandas: 2.0.0 numpy: 1.24.2 scipy: 1.10.1 sympy: 1.11.1 humanfriendly: 10.0 -------------------- date and time = 14/04/2023 13:30:21 -------------------- ---------------------------------------- ***** Statistics for L(1,chi_quad) ***** name source fileanalysis = merged-graph6-Lquad-total.result ***** MAX L(1,chi_quad) ***** maximal value on column Lquad1 = 6.3998733979784683 attained at q = 4305479 second maximal value on column Lquad1 = 6.3922747258449748 attained at q = 9067439 third minimal value on column Lquad1 = 6.3584872717961547 attained at q = 5892311 fourth minimal value on column Lquad1 = 6.3522794339242070 attained at q = 8282039 ***** MIN L(1,chi_quad) ***** minimal value on column Lquad1 = 0.1969884966155210 attained at q = 7105733 second minimal value on column Lquad1 = 0.1988143276200977 attained at q = 991027 third minimal value on column Lquad1 = 0.2024867397751592 attained at q = 3370517 fourth minimal value on column Lquad1 = 0.2046563391074435 attained at q = 2004917 ---------------------------------------- ***** Average values ***** name source fileanalysis = merged-graph6-Lquad-total.result The average value for Lquad1 is = 1.6396614666943508 there are 251836 characters with Lquad1 > (average of all characters); percentage = 37.89412228511928 -------- odd characters -------- there are 332398 odd characters over a total of 664578 characters; percentage = 50.016401385540895 there are 126012 odd characters with Lquad1 > (average of all characters); percentage = 18.96120545669583 The average value for Lquad1, odd characters, is = 1.6440488859281268 -------- even characters -------- there are 332180 even characters over a total of 664578 characters; percentage = 49.983598614459105 there are 125824 even characters with Lquad1 > (average of all characters); percentage = 18.93291682842345 The average value for Lquad1, even characters, is = 1.6352711681258985 ---------------------------------------- ***** Statistics for c1 = L(1,chi_quad) /log(q) ***** name source fileanalysis = merged-graph6-Lquad-total.result ***** MAX values c1 = L(1,chi_quad) /log(q) ***** maximal value on column c1 = 0.6267599041082923 attained at q = 23 second maximal value on column c1 = 0.6122601866526582 attained at q = 71 third minimal value on column c1 = 0.6102082422974060 attained at q = 7 fourth minimal value on column c1 = 0.5951049584465156 attained at q = 47 ***** MIN values c1 = L(1,chi_quad) /log(q) ***** minimal value on column c1 = 0.0124862668780409 attained at q = 7105733 second minimal value on column c1 = 0.0129990204435214 attained at q = 9798557 third minimal value on column c1 = 0.0130689260515964 attained at q = 8633957 fourth minimal value on column c1 = 0.0132987616924616 attained at q = 6301973 ---------------------------------------- ***** Statistics for c2 ***** name source fileanalysis = c3-c4-3-10e7.csv name source fileanalysis = merged-graph6-Lquad-total.result ***** MAX c2 = c1/(c3plusc4) ***** ----------------------------------------------- maximal value on column c2 = 0.4206022969159307 attained at q = 311 second maximal value on column c2 = 0.4178249669515177 attained at q = 479 third minimal value on column c2 = 0.4149789742638502 attained at q = 71 fourth minimal value on column c2 = 0.4015731149997521 attained at q = 1559 ----------------------------------------------- ***** MIN c2 = c1/(c3plusc4) ***** minimal value on column c2 = 0.0091904477672903 attained at q = 7105733 second minimal value on column c2 = 0.0095679232322323 attained at q = 9798557 third minimal value on column c2 = 0.0096193551768844 attained at q = 8633957 fourth minimal value on column c2 = 0.0097884452772606 attained at q = 6301973 ----------------------------------------------- ***** Littlewood bounds L1 = min(LLI) = 0.1231345245367277 attained at q = 3 second minimum: L1 = min(LLI) = 0.4435538721521973 attained at q = 5 third minimum : L1 = min(LLI) = 0.8675157625437266 attained at q = 163 fourth minimum : L1 = min(LLI) = 1.1302203128726604 attained at q = 991027 fifth minimum : L1 = min(LLI) = 1.1580285643633341 attained at q = 2477 ******** max(LLI) = 38.3973766224175 attained at q = 9067439 -------- L2 = max(ULI) = 1.8047109294350627 attained at q = 3 second maximum = 0.6590147671887095 attained at q = 4305479 third maximum = 0.6555330227419567 attained at q = 3190151 fourth maximum = 0.6499061623426937 attained at q = 5892311 fifth maximum = 0.6488965439916344 attained at q = 3068831 ******** min(ULI) = 0.02004720324962609 attained at q = 7105733 -------- ----------------------------------------------- ***** class number maximal class_number = 6343 attained at q = 9858551 minimal class_number = 1 attained at q = 7 ----------------------------------------------- ***** Joshi bounds ***** Joshi first bound ***** number of characters such that L(1)/ log log q > = e^gamma ***** there are 13036 cases first case of Joshifirst's bound = True attained at q = 3 second case of Joshifirst's bound = True attained at q = 7 third case of Joshifirst's bound = True attained at q = 71 fourth case of Joshifirst's bound = True attained at q = 191 fifth case of Joshifirst's bound = True attained at q = 239 ***** Joshis second bound ***** number of characters such that L(1) * log log q < = PI^2 / (6*e^gamma) ***** there are 14453 cases first case of Joshisecond's bound = True attained at q = 3 second case of Joshisecond's bound = True attained at q = 5 third case of Joshisecond's bound = True attained at q = 7 fourth case of Joshisecond's bound = True attained at q = 11 fifth case of Joshisecond's bound = True attained at q = 13 ----------------------------------------------- ***** Saving table of results ***** ***** in file named: table-results-10-digits.csv ***** ---------------------------------------- ********* plotting histogram Lquad1 ********* mean (Lquad1(q)) = 1.6396614666943508 standard deviation (Lquad1(q)) = 1.168265539263388 variance (Lquad1(q)) = 1.3648443702303747 total data 664578.0 max x = 6.710017643046616 min x = -0.11315574845262633 bin_width = 0.06823173391499242 num bins = 100 mean = 1.6396614666943508 mass = 45345.30926175783 warning: /usr/local/texlive/texmf-local/web2c/texmf.cnf:8: (kpathsea) Last line of file ends with \. ********* saved in histo-Lquad1.png ********* ---------------------------------------- ********* plotting histogram c1(q) ********* mean (c1(q)) = 0.10968773739032925 standard deviation (c1(q)) = 0.07887675467212063 variance (c1(q)) = 0.006221542427605903 total data 664578.0 max x = 0.657473585969805 min x = -0.01822741498347169 bin_width = 0.006757010009532767 num bins = 100 mean = 0.10968773739032925 mass = 4490.560198115267 ********* saved in histo-c1.png ********* ---------------------------------------- ********* plotting histogram c2(q) ********* mean (c2(q)) = 0.08072493197923117 standard deviation (c2(q)) = 0.05803753411143148 variance (c2(q)) = 0.003368355365735572 total data 664578.0 max x = 0.44117288937336274 min x = -0.011380144690141676 bin_width = 0.004525530340635044 num bins = 100 mean = 0.08072493197923117 mass = 3007.5679027185565 ********* saved in histo-c2.png ********* ---------------------------------------- ******** ******** Print scatter Lquad1 ******* ********* saved in Lquad1-scat.png ********* ---------------------------------------- ******** ******** Print scatter c1 ******* ********* saved in c1-scat.png ********* ---------------------------------------- ******** ******** Print scatter c1 - bounded heights ******* ********* saved in c1-scat-bounded.png ********* ---------------------------------------- ******** ******** Print scatter c2 ******* ********* saved in c2-scat.png ********* ---------------------------------------- ******** ******** Print scatter c2 - bounded heights ******* ********* saved in c2-scat-bounded.png ********* ---------------------------------------- ******** ******** Print scatter ULI ******* ********* saved in ULI-scat.png ********* ---------------------------------------- ******** ******** Print scatter LLI ******* ********* saved in LLI-scat.png ********* ---------------------------------------- ********* plotting histogram ULI(chi,q) ********* mean (ULI(chi, q)) = 0.1701388804624603 standard deviation (ULI(chi, q)) = 0.12131036129230785 variance (ULI(chi, q)) = 0.014716203756870262 total data 664577.0 max x = 0.6909631453856637 min x = -0.011901174947328085 bin_width = 0.007028643203329918 num bins = 100 mean = 0.1701388804624603 mass = 4671.074614139387 ********* saved in histo-ULI.png ********* ---------------------------------------- ********* plotting histogram LLI(chi,q) ********* mean (LLI(chi, q)) = 9.615707178476255 standard deviation (LLI(chi, q)) = 6.863974297217922 variance (LLI(chi, q)) = 47.11414315286827 total data 664541.0 max x = 40.26073443789474 min x = -0.7331375026045812 bin_width = 0.40993871940499316 num bins = 100 mean = 9.615707178476255 mass = 272421.08653211355 ********* saved in histo-ULI.png ********* ------------------------------------------ Total Execution time : 39.26 seconds ------------------------------------------ ***** End PYTHON script ***** languasc@zygalski Lquad_analysis %