diff --git a/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.log b/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.log index bef6b69..d189ffb 100644 --- a/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.log +++ b/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.25 (MiKTeX 23.10) (preloaded format=pdflatex 2023.10.29) 12 FEB 2024 16:55 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (MiKTeX 23.10) (preloaded format=pdflatex 2023.10.29) 20 FEB 2024 01:54 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -854,27 +854,27 @@ Underfull \hbox (badness 10000) in paragraph at lines 161--163 [6 <./graphs/3D_Complex_Graphs/product_of_sin/normalized_Complex_3D_product_of_ sin_HDSET_4.png> <./graphs/3D_Complex_Graphs/product_of_sin/normalized_Complex_ 3D_product_of_sin_HDSET_8.png>] -Underfull \hbox (badness 10000) in paragraph at lines 167--168 +Underfull \hbox (badness 10000) in paragraph at lines 166--167 [] -Underfull \hbox (badness 10000) in paragraph at lines 169--170 +Underfull \hbox (badness 10000) in paragraph at lines 168--169 [] -Underfull \hbox (badness 10000) in paragraph at lines 174--175 +Underfull \hbox (badness 10000) in paragraph at lines 173--174 [] -Underfull \hbox (badness 10000) in paragraph at lines 179--180 +Underfull \hbox (badness 10000) in paragraph at lines 184--185 [] [7] -Underfull \hbox (badness 10000) in paragraph at lines 196--197 +Underfull \hbox (badness 10000) in paragraph at lines 201--202 [] @@ -885,7 +885,7 @@ See the LaTeX manual or LaTeX Companion for explanation. 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[8] -<graphs/2D_Real_Graphs/prime_indicator_og_redone.png, id=53, 1084.80281pt x 133 +<graphs/2D_Real_Graphs/prime_indicator_og_redone.png, id=55, 1084.80281pt x 133 .24782pt> File: graphs/2D_Real_Graphs/prime_indicator_og_redone.png Graphic file (type pn g) <use graphs/2D_Real_Graphs/prime_indicator_og_redone.png> Package pdftex.def Info: graphs/2D_Real_Graphs/prime_indicator_og_redone.png u -sed on input line 237. +sed on input line 242. (pdftex.def) Requested size: 488.15675pt x 59.96095pt. File: graphs/2D_Real_Graphs/prime_indicator_og_redone.png Graphic file (type pn g) <use graphs/2D_Real_Graphs/prime_indicator_og_redone.png> Package pdftex.def Info: graphs/2D_Real_Graphs/prime_indicator_og_redone.png u -sed on input line 237. +sed on input line 242. (pdftex.def) Requested size: 488.15675pt x 59.96095pt. @@ -1244,40 +1244,40 @@ See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... -l.238 \hspace{8mm}\caption +l.243 \hspace{8mm}\caption {Figure 12: 2D Graph of $b(x)$ showcasing the zero... ? -Underfull \hbox (badness 10000) in paragraph at lines 237--239 +Underfull \hbox (badness 10000) in paragraph at lines 242--244 [] -Underfull \hbox (badness 10000) in paragraph at lines 240--245 +Underfull \hbox (badness 10000) in paragraph at lines 245--250 [] -Underfull \hbox (badness 10000) in paragraph at lines 240--245 +Underfull \hbox (badness 10000) in paragraph at lines 245--250 [] <graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/non_normaliz -ed_goodm_prism_1.png, id=54, 764.10469pt x 452.4403pt> +ed_goodm_prism_1.png, id=56, 764.10469pt x 452.4403pt> File: graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/non_nor malized_goodm_prism_1.png Graphic file (type png) <use graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/non_norm alized_goodm_prism_1.png> Package pdftex.def Info: graphs/3D_Complex_Graphs/product_of_product_representa -tion_of_sin/non_normalized_goodm_prism_1.png used on input line 247. +tion_of_sin/non_normalized_goodm_prism_1.png used on input line 252. 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Type H <return> for immediate help. ... -l.248 \hspace{8mm}\caption +l.253 \hspace{8mm}\caption {Figure 13: 2D Graph of $b(x+iy)$ showcasing the z... ? -Underfull \hbox (badness 10000) in paragraph at lines 247--249 +Underfull \hbox (badness 10000) in paragraph at lines 252--254 [] -Underfull \hbox (badness 10000) in paragraph at lines 250--251 +Underfull \hbox (badness 10000) in paragraph at lines 255--256 [] <graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/non_normaliz -ed_goodm_prism_8.png, id=55, 1407.00656pt x 727.96968pt> +ed_goodm_prism_8.png, id=57, 1407.00656pt x 727.96968pt> File: graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/non_nor malized_goodm_prism_8.png Graphic file (type png) <use graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/non_norm alized_goodm_prism_8.png> Package pdftex.def Info: graphs/3D_Complex_Graphs/product_of_product_representa -tion_of_sin/non_normalized_goodm_prism_8.png used on input line 253. +tion_of_sin/non_normalized_goodm_prism_8.png used on input line 258. 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Type H <return> for immediate help. ... -l.254 \hspace{8mm}\caption +l.259 \hspace{8mm}\caption {Figure 14: 3D Graph of $b(x+iy)$ showcasing the z... ? -Underfull \hbox (badness 10000) in paragraph at lines 253--255 +Underfull \hbox (badness 10000) in paragraph at lines 258--260 [] -Underfull \hbox (badness 10000) in paragraph at lines 256--257 +Underfull \hbox (badness 10000) in paragraph at lines 261--262 [] <graphs/2D_Complex_Graphs/Infinite_Product_of_infinite_product_representation_o -f_sin/Complex_product_17_n_2-28__imaginary_scalar_logarithm.png, id=59, 1354.30 +f_sin/Complex_product_17_n_2-28__imaginary_scalar_logarithm.png, id=61, 1354.30 97pt x 720.44156pt> File: graphs/2D_Complex_Graphs/Infinite_Product_of_infinite_product_representat ion_of_sin/Complex_product_17_n_2-28__imaginary_scalar_logarithm.png Graphic fi @@ -1350,7 +1350,7 @@ le (type png) on_of_sin/Complex_product_17_n_2-28__imaginary_scalar_logarithm.png> Package pdftex.def Info: graphs/2D_Complex_Graphs/Infinite_Product_of_infinite_ product_representation_of_sin/Complex_product_17_n_2-28__imaginary_scalar_logar -ithm.png used on input line 260. +ithm.png used on input line 265. 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Type H <return> for immediate help. ... -l.261 \hspace{8mm}\caption +l.266 \hspace{8mm}\caption {Figure 15: 2D Fractal Colorization complex plot o... ? -Underfull \hbox (badness 10000) in paragraph at lines 260--262 +Underfull \hbox (badness 10000) in paragraph at lines 265--267 [] -Underfull \hbox (badness 10000) in paragraph at lines 263--264 +Underfull \hbox (badness 10000) in paragraph at lines 268--269 [] @@ -1386,17 +1386,17 @@ Underfull \hbox (badness 10000) in paragraph at lines 263--264 rmalized_goodm_prism_8.png> <./graphs/2D_Complex_Graphs/Infinite_Product_of_inf inite_product_representation_of_sin/Complex_product_17_n_2-28__imaginary_scalar _logarithm.png>] -<graphs/2D_Real_Graphs/prime_indicator_norm.png, id=63, 1349.79282pt x 191.9672 +<graphs/2D_Real_Graphs/prime_indicator_norm.png, id=65, 1349.79282pt x 191.9672 pt> File: graphs/2D_Real_Graphs/prime_indicator_norm.png Graphic file (type png) <use graphs/2D_Real_Graphs/prime_indicator_norm.png> Package pdftex.def Info: graphs/2D_Real_Graphs/prime_indicator_norm.png used o -n input line 275. +n input line 280. 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Type H <return> for immediate help. ... -l.276 \hspace{8mm}\caption +l.281 \hspace{8mm}\caption {Figure 16: 2D Graph of $B(x)$ showcasing the zero... ? -Underfull \hbox (badness 10000) in paragraph at lines 275--277 +Underfull \hbox (badness 10000) in paragraph at lines 280--282 [] -Underfull \hbox (badness 10000) in paragraph at lines 278--279 +Underfull \hbox (badness 10000) in paragraph at lines 283--284 [] -Underfull \hbox (badness 10000) in paragraph at lines 280--281 +Underfull \hbox (badness 10000) in paragraph at lines 285--286 [] <graphs/3D_Complex_Graphs/product_of_product_representation_of_sin//ComplexPlot -_prodprodforsin_4_Prism_Colorization_New.png, id=64, 1390.44469pt x 723.45282pt +_prodprodforsin_4_Prism_Colorization_New.png, id=66, 1390.44469pt x 723.45282pt > File: graphs/3D_Complex_Graphs/product_of_product_representation_of_sin//Comple xPlot_prodprodforsin_4_Prism_Colorization_New.png Graphic file (type png) @@ -1434,7 +1434,7 @@ xPlot_prodprodforsin_4_Prism_Colorization_New.png Graphic file (type png) Plot_prodprodforsin_4_Prism_Colorization_New.png> Package pdftex.def Info: graphs/3D_Complex_Graphs/product_of_product_representa tion_of_sin//ComplexPlot_prodprodforsin_4_Prism_Colorization_New.png used on i -nput line 284. +nput line 289. 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Type H <return> for immediate help. ... -l.285 \hspace{8mm}\caption +l.290 \hspace{8mm}\caption {Figure 17: 2D Graph of $B(x+iy)$ showcasing the z... ? -Underfull \hbox (badness 10000) in paragraph at lines 284--286 +Underfull \hbox (badness 10000) in paragraph at lines 289--291 [] <graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/ComplexPlot_ -prodprodforsin_8.png, id=65, 791.20593pt x 586.44093pt> +prodprodforsin_8.png, id=67, 791.20593pt x 586.44093pt> File: graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/Complex Plot_prodprodforsin_8.png Graphic file (type png) <use graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/ComplexP lot_prodprodforsin_8.png> Package pdftex.def Info: graphs/3D_Complex_Graphs/product_of_product_representa -tion_of_sin/ComplexPlot_prodprodforsin_8.png used on input line 291. +tion_of_sin/ComplexPlot_prodprodforsin_8.png used on input line 296. (pdftex.def) Requested size: 356.03938pt x 263.89597pt. 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Type H <return> for immediate help. ... -l.292 \hspace{8mm}\caption +l.297 \hspace{8mm}\caption {Figure 18: 3D Graph of $B(x+iy)$ showcasing the z... ? -Underfull \hbox (badness 10000) in paragraph at lines 291--293 +Underfull \hbox (badness 10000) in paragraph at lines 296--298 [] <graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/ComplexPlot_ -norm_prodprodforsin_prism_3.png, id=71, 934.24031pt x 646.66594pt> +norm_prodprodforsin_prism_3.png, id=72, 934.24031pt x 646.66594pt> File: graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/Complex Plot_norm_prodprodforsin_prism_3.png Graphic file (type png) <use graphs/3D_Complex_Graphs/product_of_product_representation_of_sin/ComplexP lot_norm_prodprodforsin_prism_3.png> Package pdftex.def Info: graphs/3D_Complex_Graphs/product_of_product_representa -tion_of_sin/ComplexPlot_norm_prodprodforsin_prism_3.png used on input line 296 +tion_of_sin/ComplexPlot_norm_prodprodforsin_prism_3.png used on input line 301 . 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Missing } inserted. <inserted text> } -l.571 \end{align*} +l.576 \end{align*} ? ! Double superscript. <argument> ...1}{\Gamma \left (z\right )} &= ze^z^ \gamma \prod _{n=2}^z\left... -l.571 \end{align*} +l.576 \end{align*} ? ! Missing { inserted. <to be read again> \left -l.571 \end{align*} +l.576 \end{align*} ? ! Missing } inserted. <inserted text> } -l.571 \end{align*} +l.576 \end{align*} ? ! Double superscript. <argument> ... \frac {\pi }{z} \left (\left [ze^z^ \gamma \prod _{n=2}^z \lef... -l.583 \end{align*} +l.588 \end{align*} ? ! Double superscript. <argument> ... \frac {\pi }{z} \left (\left [ze^z^ \gamma \prod _{n=2}^z \lef... -l.583 \end{align*} +l.588 \end{align*} ? [23] @@ -1860,13 +1860,13 @@ File: graphs/2D_Real_New/Prime_Indicator_H_normalized_labeled.png Graphic file (type png) <use graphs/2D_Real_New/Prime_Indicator_H_normalized_labeled.png> Package pdftex.def Info: graphs/2D_Real_New/Prime_Indicator_H_normalized_labele -d.png used on input line 595. +d.png used on input line 600. (pdftex.def) Requested size: 411.26465pt x 94.17755pt. File: graphs/2D_Real_New/Prime_Indicator_H_normalized_labeled.png Graphic file (type png) <use graphs/2D_Real_New/Prime_Indicator_H_normalized_labeled.png> Package pdftex.def Info: graphs/2D_Real_New/Prime_Indicator_H_normalized_labele -d.png used on input line 595. +d.png used on input line 600. (pdftex.def) Requested size: 411.26465pt x 94.17755pt. @@ -1876,12 +1876,12 @@ See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... -l.596 \hspace{8mm}\caption +l.601 \hspace{8mm}\caption {Figure 39: 2D Graph of $J(x)$ and $H(x)$ showcasi... ? -Underfull \hbox (badness 10000) in paragraph at lines 595--597 +Underfull \hbox (badness 10000) in paragraph at lines 600--602 [] @@ -1892,13 +1892,13 @@ File: graphs/2D_Real_New/Prime_Indicator_J_normalized.png Graphic file (type pn g) <use graphs/2D_Real_New/Prime_Indicator_J_normalized.png> Package pdftex.def Info: graphs/2D_Real_New/Prime_Indicator_J_normalized.png u -sed on input line 636. +sed on input line 641. (pdftex.def) Requested size: 386.87332pt x 113.82613pt. File: graphs/2D_Real_New/Prime_Indicator_J_normalized.png Graphic file (type pn g) <use graphs/2D_Real_New/Prime_Indicator_J_normalized.png> Package pdftex.def Info: graphs/2D_Real_New/Prime_Indicator_J_normalized.png u -sed on input line 636. +sed on input line 641. (pdftex.def) Requested size: 386.87332pt x 113.82613pt. @@ -1908,26 +1908,26 @@ See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... -l.637 \hspace{8mm}\caption +l.642 \hspace{8mm}\caption {Figure 40: 2D Graph of $J_1(x)$ showcasing the no... ? -Underfull \hbox (badness 10000) in paragraph at lines 636--638 +Underfull \hbox (badness 10000) in paragraph at lines 641--643 [] ! Missing { inserted. <to be read again> \left -l.640 ...of a(z), we have all $0^0=1$ and $0^\left +l.645 ...of a(z), we have all $0^0=1$ and $0^\left (nonzero\right)$ $=0$, thi... ? ! Missing } inserted. <inserted text> } -l.640 ... all $0^0=1$ and $0^\left(nonzero\right)$ +l.645 ... all $0^0=1$ and $0^\left(nonzero\right)$ $=0$, this forced all pri... ? @@ -1937,13 +1937,13 @@ File: graphs/2D_Real_New/Composite_Indicator_H_normalized.png Graphic file (typ e png) <use graphs/2D_Real_New/Composite_Indicator_H_normalized.png> Package pdftex.def Info: graphs/2D_Real_New/Composite_Indicator_H_normalized.pn -g used on input line 648. +g used on input line 653. (pdftex.def) Requested size: 405.6185pt x 160.12445pt. File: graphs/2D_Real_New/Composite_Indicator_H_normalized.png Graphic file (typ e png) <use graphs/2D_Real_New/Composite_Indicator_H_normalized.png> Package pdftex.def Info: graphs/2D_Real_New/Composite_Indicator_H_normalized.pn -g used on input line 648. +g used on input line 653. (pdftex.def) Requested size: 405.6185pt x 160.12445pt. ! LaTeX Error: \caption outside float. @@ -1952,12 +1952,12 @@ See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... -l.649 \hspace{8mm}\caption +l.654 \hspace{8mm}\caption {Figure 41: 2D Graph of $J_2(x)$ showcasing the no... ? -Underfull \hbox (badness 10000) in paragraph at lines 648--650 +Underfull \hbox (badness 10000) in paragraph at lines 653--655 [] @@ -1966,25 +1966,25 @@ ew/Composite_Indicator_H_normalized.png>] ! Misplaced alignment tab character &. <recently read> & -l.653 ...tion for L(z), M(z), N(z), O(z), P(z), $& +l.658 ...tion for L(z), M(z), N(z), O(z), P(z), $& $ Q(z) resemble the leadin... ? [26] [27] ! Misplaced alignment tab character &. -l.747 ...come commands the rule the $(2)^{0}=(1) & +l.752 ...come commands the rule the $(2)^{0}=(1) & (2)^{1}=(2)$ such that it... ? ! Misplaced alignment tab character &. <recently read> & -l.747 ...o command the rule that $(-1)^{2}=(1)$ $& +l.752 ...o command the rule that $(-1)^{2}=(1)$ $& $ $(-1)^{1}=(-1)$, combini... ? ! Misplaced alignment tab character &. -l.758 ...mately executing the rule, $(2)^{0}=(1) & +l.763 ...mately executing the rule, $(2)^{0}=(1) & (2)^{1}=(2)$, this can be... ? @@ -1992,70 +1992,52 @@ l.758 ...mately executing the rule, $(2)^{0}=(1) & ! Missing $ inserted. <inserted text> $ -l.773 ... representation of the dirichlet \eta(z)} +l.778 ... representation of the dirichlet \eta(z)} ? ! Missing $ inserted. <inserted text> $ -l.773 ... representation of the dirichlet \eta(z)} +l.778 ... representation of the dirichlet \eta(z)} ? -<graphs/R1_actual_BIG.png, id=166, 1808.25563pt x 514.17094pt> -File: graphs/R1_actual_BIG.png Graphic file (type png) -<use graphs/R1_actual_BIG.png> -Package pdftex.def Info: graphs/R1_actual_BIG.png used on input line 777. -(pdftex.def) Requested size: 470.13538pt x 133.68129pt. -File: graphs/R1_actual_BIG.png Graphic file (type png) -<use graphs/R1_actual_BIG.png> -Package pdftex.def Info: graphs/R1_actual_BIG.png used on input line 777. -(pdftex.def) Requested size: 470.13538pt x 133.68129pt. - -Overfull \hbox (69.56842pt too wide) detected at line 781 + +Overfull \hbox (69.56842pt too wide) detected at line 787 [] [] -[29 <./graphs/R1_actual_BIG.png>] +[29] ! Missing $ inserted. <inserted text> $ -l.808 ...n rate of $S_1(z)$ resembles that of \eta +l.814 ...n rate of $S_1(z)$ resembles that of \eta (s), ? ! Missing $ inserted. <inserted text> $ -l.809 +l.815 ? -<graphs/S1_actual_BIG.png, id=171, 1877.51437pt x 517.18219pt> -File: graphs/S1_actual_BIG.png Graphic file (type png) -<use graphs/S1_actual_BIG.png> -Package pdftex.def Info: graphs/S1_actual_BIG.png used on input line 812. -(pdftex.def) Requested size: 488.14223pt x 134.46419pt. -File: graphs/S1_actual_BIG.png Graphic file (type png) -<use graphs/S1_actual_BIG.png> -Package pdftex.def Info: graphs/S1_actual_BIG.png used on input line 812. -(pdftex.def) Requested size: 488.14223pt x 134.46419pt. - -Overfull \hbox (68.10747pt too wide) detected at line 818 + +Overfull \hbox (68.10747pt too wide) detected at line 824 [] [] ! Missing $ inserted. <inserted text> $ -l.842 if $S_1(s)$ is Holomorphic with \eta +l.848 if $S_1(s)$ is Holomorphic with \eta (s). ? ! Missing $ inserted. <inserted text> $ -l.843 +l.849 ? -[30 <./graphs/S1_actual_BIG.png>] -Overfull \hbox (92.06665pt too wide) in paragraph at lines 847--847 +[30] +Overfull \hbox (92.06665pt too wide) in paragraph at lines 853--853 \OT1/cmtt/m/n/10 courses / 18 -[] 104 -[] seminar -[] in -[] analysis -[] appli cations -[] to -[] number -[] theory -[] fall -[] 2006 / 58778de0faae6448c3b0bf b1a1ec4f7a _ @@ -2073,10 +2055,10 @@ ml'. ) Here is how much of TeX's memory you used: - 11063 strings out of 475248 - 231587 string characters out of 5758756 - 1938785 words of memory out of 5000000 - 32508 multiletter control sequences out of 15000+600000 + 11041 strings out of 475248 + 230931 string characters out of 5758756 + 1940785 words of memory out of 5000000 + 32486 multiletter control sequences out of 15000+600000 563959 words of font info for 57 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 66i,17n,81p,926b,1271s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -2102,9 +2084,9 @@ type1/public/amsfonts/cm/cmsy7.pfb><C:/Users/ADA/AppData/Local/Programs/MiKTeX/ fonts/type1/public/amsfonts/cm/cmti10.pfb><C:/Users/ADA/AppData/Local/Programs/ MiKTeX/fonts/type1/public/amsfonts/cm/cmtt10.pfb> Output written on Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis. -pdf (31 pages, 14866303 bytes). +pdf (31 pages, 14732928 bytes). PDF statistics: - 261 PDF objects out of 1000 (max. 8388607) + 257 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) - 216 words of extra memory for PDF output out of 10000 (max. 10000000) + 206 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.pdf b/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.pdf index 2bfc0f6..e85abc8 100644 Binary files a/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.pdf and b/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.pdf differ diff --git a/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.tex b/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.tex index c9a9eac..b9a32e4 100644 --- a/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.tex +++ b/Divisor_Waves_and_their_Connection_to_the_Riemann_Hypothesis.tex @@ -161,7 +161,6 @@ \subsection*{III. Initial Description of Function A(z)} \end{align*} \hspace{8mm}\caption{Figure 11: 2D Graph of $A(x+iy)$ showcasing the normalized zeros of the real numbers in the complex plane} \\ - \newpage \subsection*{IV. Derivation of a(z) and b(z) from the Weierstrass product formula for sin} Derivation of a(z) and b(z) by substituting the infinite product representation of $sin(\pi*z/n)$ into the infinite product of $\sin(\pi*z/n)$: \\ @@ -176,7 +175,13 @@ \subsection*{IV. Derivation of a(z) and b(z) from the Weierstrass product formul \prod_{k=2}^x\sin\left(\frac{\pi z}{k}\right) = \prod_{k=2}^x \pi z\prod_{n=2}^x \left(1-\frac{z^2}{n^2k^2}\right) \\ \end{align*} -This proof shows the relationship between a(z) and b(z), and allows us to appreciate some of the beautiful relationships between these functions. The function b(z) is an equivalent form of a(z) however it has extra parameters that we can use to our advantage. \\ +and for use later we will optimize the upper index values for n and k to reduce the number of product expansions in the calculation as: + +\begin{align*} + b(z) = \prod_{k=2}^{\sqrt{x}} \pi z\prod_{n=2}^{\frac{x}{2}} \left(1-\frac{z^2}{n^2k^2}\right) \\ +\end{align*} + +where k now goes to $\sqrt{x}$ and n goes to $\frac{x}{2}$. This is the minimum value which still allows the sieve to catch all factors of z in b(z). This proof shows the relationship between a(z) and b(z), and allows us to appreciate some of the beautiful relationships between these functions. The function b(z) is an equivalent form of a(z) however it has extra parameters that we can use to our advantage. \\ Now, we can expand the product out to infinity, using the expression we derived earlier: @@ -557,7 +562,7 @@ \subsection*{XV. Gamma and Sin Product Identities} \begin{align*} \prod_{n=2}^z\left[sin(\frac{\pi * z}{n})\right] &= \prod_{n=2}^z\left[\frac{\pi}{\Gamma\left(\frac{z}{n}\right) * \Gamma\left(1 - \frac{z}{n}\right)}\right] \\ \end{align*} -now taking a look at the original euler infinite product representation of sin, +now taking a look at the original Weierstrass infinite product representation of sin, \begin{align*} sin(\pi z) &= z\pi * \prod_{k=2}^z\left[1 - \frac{z^2}{k^2}\right] \\ \end{align*} @@ -771,9 +776,10 @@ \subsection*{XVIII. Multi-variable Combinatoric Alternating Sequence L(z) for Pr \newpage \subsection*{XVIV. Divisor Wave Holomorphic representation of the Euler Product representation of the dirichlet \eta(z)} -Continuing where we left off, the formulae $J(z)$, and $M(z)$ will be combined to showcase an error approximation between $J_1(z)$ and the dirichlet $\eta(s)$ function. Then in the next example, we will utilize $Q_1(z)$ to create a Holomorphic representation of $\eta(s)$ called $S_1(z)$ +Continuing where we left off, the formulae $J_1(z)$, and $M(z)$ will be combined to showcase an error approximation between $J_1(z)$ and the dirichlet $\eta(s)$ function. Then in the next example, we will utilize $Q_1(z)$ to create a Holomorphic representation of $\eta(s)$ called $S_1(z)$ + \begin{align*} -\includegraphics[scale=0.26]{graphs/R1_actual_BIG} +R_1(z) = \prod_{n=2}^x \left( \frac{1}{1 + M_1(z) * J_1(z)} \right) \end{align*} \begin{align*} @@ -808,7 +814,7 @@ \subsection*{XVIV. Divisor Wave Holomorphic representation of the Euler Product The Function $S_1(z)$ utilizes the alternator $Q_1(z)$ as well as $J(z)$ which constrain the linear relationship of the prime indicating function, to the alternation rate of $\eta(s)$. $S_1(z)$ ultimately connects the zeros and the non-zeros of the primes to the dirichlet eta function, the function J(z) is defined such that these zeros all lie on the line $y = x$, as a result, the zeros of the zeta functions must all lie on the critical strip. After transformation as an input to an output, these values hold the relationship $y=x$ and upon translation $z = 1/2 + iy$ contains all non-trivial zeros of zeta. From the expansion we can see that the alternation rate of $S_1(z)$ resembles that of \eta(s), \begin{align*} -\includegraphics[scale=0.26]{graphs/S1_actual_BIG} +S_1(z) = \prod_{n=2}^x \left( \frac{1}{1 + Q_1(z) * J_1(z)} \right) \end{align*} expanding $S_1(z)$ like $R_1(z)$, we see a similar reduction however this time the alternator function $Q_1(z)$ follows the prime and composite numbers, as outlined by the expansion of $Q_1(z)$ in the previous Chapter. diff --git a/graphs/2D_Real_New/code_optimization.png b/graphs/2D_Real_New/code_optimization.png new file mode 100644 index 0000000..6cf43b3 Binary files /dev/null and b/graphs/2D_Real_New/code_optimization.png differ