$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions

Solution 1:

(1) is a correct computation. In general, to treat primes of the form $kn+m$, you would have a linear combination of $\phi(k)$ sums, each of which runs over the zeros of a different Dirichlet $L$-function (of which the Riemann $\zeta$ function is a special case). And yes, assuming the generalized Riemann hypothesis, all of the terms including those sums over zeros can be estimated into the $O(x^{1/2+\epsilon})$ term.

To find out more, you want to look for "the prime number theorem for arithmetic progressions", and in particular the "explicit formula". I know it appears in Montgomery and Vaughan's book, for example.

Solution 2:

Here is a table of the 331 first zeros for numerical evaluation (warning: values could be missing...)

6.0209489046975966549025115255,
10.243770304166554552137757473,
12.988098012312422507453109785,
16.342607104587222194976861486,
18.291993196123534838526004279,
21.450611343983460497200948384,
23.278376520459531531819558882,
25.728756425088727567265088675,
28.359634343025327785651607948,
29.656384014593152721809906963,
32.592186527117155130815194045,
34.199957509213146913044795479,
36.142880458303137830565814472,
38.511923141718691293776504680,
40.322674066690544180344394367,
41.807084620004562337157521896,
44.617891058662303393482045721,
45.599584396791566745937702295,
47.741562280939141250781347344,
49.723129323782586066569570883,
51.686093452870528439533811111,
52.768820767804729265035076579,
55.267543584699224846718259652,
56.934374055202296886801711960,
58.116707110673917977262367546,
60.421713949007834673018618295,
62.008632285767769451930615509,
63.714641118785433123518297959,
64.976170573095999348605924739,
67.636920863546068398054991150,
68.365884503834422961233738148,
70.185879908802112061371282120,
72.155484974381881214691542592,
73.767635521485893336154626169,
75.143121647433111405798000424,
76.696303203430199064566659967,
78.809998314320913003691783203,
80.210131238366638915148032664,
81.213951626883151157734833354,
83.666656014470571651283310367,
84.731740363781628608217577601,
86.577660168390264410210548722,
87.629718119587899689039432381,
89.801131616695811325969388292,
91.349703814697573473931013977,
92.237499910454258046004175773,
94.166619585960021307053060933,
96.136011161780558185274479276,
96.961741579417483577607743327,
98.755300415754527668603973554,
100.13488670306768529019231904,
102.14138082688961382675997843,
103.28807538167900270159333920,
104.33326984426745450495085217,
106.69445890889584172960040021,
107.69020697514670020812946699,
110.49960817642909478048490816,
112.36781674401217995716661013,
113.81479554899267353956309211,
116.19320465846121898963013972,
118.53755437884686165780011347,
119.45298987620909035824186237,
120.73129361866037693368487739,
122.44746137906871191967947661,
123.79454876031950609879380618,
125.76851955991559376249639951,
126.29877602494140909790014712,
127.95940768306329938305588877,
129.88562335864546932623503678,
131.09357875408399607523436803,
132.14357660098782788050132120,
133.74418146397481319142376707,
135.49083725255700224755404510,
136.54731226728255838933127345,
138.45729450969625766509038076,
138.75017770461144012954768949,
141.25363250975178412806669861,
142.39441752219778229834700120,
143.32906274161773201159359077,
144.97816627771136218178674826,
146.52200528490833848995838351,
147.93453081218091703481025120,
150.29635900154312145801640888,
151.96198767048369072791846343,
153.69961262351526052550597524,
154.57549142871991733629296692,
155.65024866324343586811534637,
157.74830530790292831907457397,
158.70502112552935112493067703,
160.23648409677104233377768408,
161.40714697615670734528135116,
162.56604668959903962627418377,
164.73116461657689691990755216,
165.40141928586315409461482322,
166.75387916910842711467929232,
168.04442078170974462074284599,
170.05113118752669021484810142,
170.73476699457880545645576325,
172.28048177162133811838196765,
173.44297883613501571697079678,
174.91508808540057340441548453,
176.59730214191806744299558376,
177.70121257457377580802426815,
178.36237490898565560941206095,
180.56931038528553139779449794,
181.61491373170438998090812364,
182.91676832896896722720668364,
184.11503237536838876567103733,
185.37399660770047491721481058,
187.06876059069440954145591925,
188.27137433227075068590376587,
189.49173149212854251358185449,
190.37118761075577274333641876,
192.36113787605631840741605201,
193.79707292614668099455436082,
194.23188322936089922195285019,
196.13200565468428482335296831,
197.11344622157640853776338155,
198.80647573614690671134916226,
200.16203942869863020720223578,
200.90161787076887940260335396,
202.26057799593173083976534604,
204.22107078015820641522076416,
204.99200345509828370916637889,
206.41191104761124331431101684,
207.31737748188785830627423332,
209.22775249264801305239596646,
210.10318551739854017072638068,
211.83341182042389919211672672,
212.53760753721342419767373639,
213.76184210127810212713458895,
215.79381838201559960577042888,
216.70341526620296547634366018,
217.58193141803801709611175843,
220.40562171063697515795993634,
221.92854850725447804914936995,
223.00310318062916135528239779,
224.12223848794683215004758755,
225.29326147520535806546186025,
226.98818848837537471961421291,
228.40550629743637922559802583,
228.95902363267523675101910098,
230.33013057780480230594069311,
232.10048174706901195203683135,
233.04806374529880402510570392,
234.35178770183278559126691643,
235.83621877114146878871598106,
236.24156049447563754765718358,
238.53711304998262144198336571,
239.33848813561967049191160837,
240.62671116939306732194741836,
241.47792092821119586131445270,
243.22893117973858440259094571,
244.51316858358815891067982334,
245.56488138011395915721891054,
246.72405853472513520671442287,
247.99511858273842666345737680,
249.18450896488545637926359681,
251.08656315045619512213120503,
251.63691583961430554302515677,
252.62344577966837570410866022,
254.31443735012273900901544828,
255.83791804974205189835948383,
256.50458494734840573390589089,
258.16526285035121855236577743,
258.83447052352496550977190962,
260.43047213492557515755933298,
261.91361498764980573464503310,
262.88361734246063150509166818,
264.05425788812825390396163665,
264.93285914122008364534327761,
267.00065119971523220684196260,
267.80144698460232962989982417,
268.78330606204792631747966001,
270.27808716748190417041357469,
271.25498369444883796265227849,
272.75587860169341823184636255,
274.17145831251909085018642460,
275.03310123658021403265232025,
275.85894601108307686333373631,
277.77232263831925417841397013,
278.80368388728653858809887697,
280.15714487097460512335476030,
280.79310694940911465165921265,
282.37964046356129969375983170,
283.60454343388307793996827161,
284.92559582341671675890997136,
286.08023330771695304124563007,
287.14942464824185193072788536,
287.97847626332132927976356976,
290.25214699517454327999080838,
290.67729632326071525164544826,
291.83202458884475548513071512,
293.20243417747287711021312313,
294.32726845145976737666041017,
295.80347082471925776988952183,
296.90066613885493288162667737,
298.07887411489226009930190937,
298.87032030707756002676910778,
300.43445985053453461820420384,
301.91984252859711814323532054,
302.91793758311719513681600129,
303.66111804033389116484131301,
305.06520886541900440815050046,
306.79513396371212102425959963,
307.28608493671484668789621993,
309.12248303042252096412050493,
309.73990761397483015859790586,
310.82367496094542226066492277,
312.55636686756100688064634699,
313.87794245506793787784240517,
314.43316516928890292103837772,
315.73498612492696188725190719,
317.01231804223259726515461799,
318.45621725635502024358202211,
319.57969129703539985620862417,
320.40630364449298983627652971,
322.00196780449032992290146609,
322.53911742379191664486892560,
324.51844467372429468247621743,
325.47478404410633875373192686,
326.45854121986539504473910264,
327.25993176651157705164654818,
329.20009793757918038019912741,
330.03604959034586362558252897,
331.21140289958185800530565896,
332.57294652823343974554556525,
333.18150832869444693922851618,
334.76869989467941751441968305,
336.15271755443328443350210094,
337.13966305477713627524956702,
338.22628268603666690396333911,
339.01125261519977890816451524,
340.84195235235292162066710614,
342.02615318336469741374106741,
342.68585823735877067928271577,
344.0836661565686862627004762,
345.2097753413459573044110626,
346.2627604671016941476647978,
347.9246481143632697374566129,
348.9817983885468120149499228,
349.4343424566002814630006007,
350.9772131093086171005228543,
352.3917682431248441639303136,
353.6933333256829546124685137,
354.3017377414641314355505898,
355.7444402625970306431358347,
356.6276472137673985753141096,
358.2882750929424366211009481,
359.0964611926055643566214138,
360.7110410245589243479893582,
361.1974930362084806074524790,
362.2769688931667497705396854,
364.4197244318352155945317356,
364.8037360921119966497987857,
366.0496895182690012543463580,
367.1255349350747462262053663,
368.4338148898077761672916077,
369.5016933318964405363710183,
371.0690160905648737699591025,
371.7700959151502466275535791,
372.8848608865911226393246899,
373.9348689154616523799502265,
375.5468747270075926960247017,
376.6839009379746539367477905,
377.5592232741848044307652399,
378.3869290457616630146695094,
380.0570731823139939966249588,
381.0486701230352553532850229,
382.2397664190157338254422377,
383.4850923542027975470084695,
384.3728456816805919723012968,
385.2152276718100110520672671,
387.1492092876308094595627758,
388.0821424777153497850755292,
388.7777179618526297788291030,
390.1691126288973758248577388,
391.0819257199607056364820762,
392.8517629133617494151722895,
393.4434077110107140920915261,
394.8199970917935633992004384,
395.8046096034178296276558608,
396.7344554225485063080197069,
398.2504159767379466957124019,
399.5655109221428371501170699,
400.6928757687115794343838349,
400.8264309422547318135722239,
402.8183184588996972949913858,
403.9074050434836080884771320,
404.9455862206118613759957847,
406.1612477978882182837592234,
407.1450155618871805968322187,
408.0395266729539842871170633,
409.5882131208357647653279288,
410.8716628634833972562447402,
411.6605714508530686255969330,
412.7311469575827245352595916,
413.6692853952956581370774306,
415.4292958729206053433345755,
416.3705912266159986762147974,
417.1292702136133141970538904,
418.5574767268274943568745235,
419.4874594223070529595380503,
420.5289538979833121243014068,
422.5103730295799395812912385,
422.6872546051342268536572665,
424.0668954927375279116779846,
424.8233888200049631864585039,
426.7324170086494599710725394,
427.3840178113418017106211713,
428.6981079540674684462150564,
429.5893859724852216637696316,
430.6175578358959205969367714,
432.0607725159701751635973864,
433.0374431195775491061980546,
434.4393523989532303725779273,
435.3481541603433508857855094,
435.9041636774496664998333698,
437.5822193991931840771232678,
439.1581082551973744286998591,
439.5375694451764723240134399,
440.6772278998545002559418857,
442.1697399171824047073380865,
442.7770442912218304593990887,
444.4382905078887461488730008,
445.4637391684133745717113130,
446.4483316219370803911040887,
447.3020089048109206937709813,
448.5069310886578395843178830,
450.0566203069175123775802095,
451.0707759054247023168568904

Solution 3:

Here is how far I got with an explicit formula for the number of primes of the form $4n+3$ below $x$, $\pi^*(x;4,3)$, expressed in terms of (sums of) sums of Riemann's $R$ functions over roots of Riemann's $\zeta$ resp. Dirichlet $\beta$ function:

\begin{align*} \Pi^*(x;4,3) &= \pi^*(x;4,3) + \tfrac12 \sum_{\substack{b\pmod 4 \\ b^2\equiv 3\pmod 4}} \pi^*(x^{1/2};4,b) + \tfrac13 \sum_{\substack{c\pmod q \\ c^3\equiv 3\pmod 4}} \pi^*(x^{1/3};4,c) + \cdots \\ \end{align*} Then I try to complete things by adding several up

\begin{align*} \Pi^*(x;4,3) &= \tfrac11\pi^*(x;4,3) + \tfrac13 \pi^*(x^{1/3};4,3) + \cdots \\ \tfrac12\Pi^*(x^{1/2};4,3) &= \tfrac12\pi^*(x^{1/2};4,3) + \tfrac16 \pi^*(x^{1/6};4,3) + \cdots \\ \tfrac14\Pi^*(x^{1/4};4,3) &= \tfrac14\pi^*(x^{1/4};4,3) + \tfrac1{12} \pi^*(x^{1/12};4,3) + \cdots \\ &\vdots&\\ \hline\\ \tag{1}\sum_{k=0}^\infty 2^{-k}\Pi^*(x^{2^{-k}};4,3)&=\sum_{m=1}^\infty \tfrac1m \pi^*(x^{1/m};4,3) \end{align*} Using Möbuis inversion I'll get

\begin{align*} \pi^*(x;4,3)&=\sum_{m=1}^\infty \tfrac{\mu(m)}m\sum_{k=0}^\infty 2^{-k}\Pi^*(x^{2^{-k}/m};4,3)\\ \tag{2}&=\sum_{k=0}^\infty 2^{-k}\sum_{m=0}^\infty \tfrac{\mu(m)}m\Pi^*(x^{2^{-k}/m};4,3) \end{align*} Now I use

\begin{align*} \Pi^*(x^{2^{-k}};4,3)&=\frac1{\phi(4)} \sum_{\chi\pmod 4} \overline{\chi(3)}\Pi^*(x^{2^{-k}},\chi)\\ \tag{3}&=\frac12 \left( \Pi^*(x^{2^{-k}},\chi_1)- \Pi^*(x^{2^{-k}},\chi_2) \right) \end{align*} and then

\begin{align*} \tag{$4_1$}\Pi^*(x^{2^{-k}},\chi_k)&=\operatorname{li}(x^{1/2^{k}})-\sum_{\rho_\zeta} \operatorname{li}(x^{\rho_\zeta/2^k})\text{ if $k=1$}\\ \tag{$4_2$}&=\phantom{\operatorname{li}(x^{1/2^{k}})}-\sum_{\rho_\beta} \operatorname{li}(x^{\rho_\beta/2^k})\text{ if $k=2$}\\ \end{align*} which gives

\begin{align*} \tag{3'}\Pi^*(x^{2^{-k}};4,3)&=\frac12 \left( \operatorname{li}(x^{1/2^{k}})-\sum_{\rho_\zeta} \operatorname{li}(x^{\rho_\zeta/2^k}) +\sum_{\rho_\beta} \operatorname{li}(x^{\rho_\beta/2^k}) \right) \end{align*} so finally

\begin{align*} \pi^*(x;4,3)&=\sum_{k=0}^\infty 2^{-k}\sum_{m=0}^\infty \tfrac{\mu(m)}m\frac12 \left( \operatorname{li}(x^{1/2^{k}})-\sum_{\rho_\zeta} \operatorname{li}(x^{\rho_\zeta/2^k}) +\sum_{\rho_\beta} \operatorname{li}(x^{\rho_\beta/2^k}) \right)\\ \tag{5}&=\sum_{k=0}^\infty 2^{-k-1}\left( \operatorname{R}(x^{1/2^{k}})-\sum_{\rho_\zeta} \operatorname{R}(x^{\rho_\zeta/2^k}) +\sum_{\rho_\beta} \operatorname{R}(x^{\rho_\beta/2^k}) \right) \end{align*}

I would be very, very glad to read your opinion...