Is there a better upper bound for the primorial $x\#$ than $4^x$
For a fairly pleasant version, for $x>1,$ we have $$ \log(x \#) < x \left(1 + \frac{1}{2 \log x} \right). $$ This is Theorem 4, formula 3.15 in Rosser and Schoenfeld (1962), Illinois J. Math.
Then, for $x \geq 563,$ we have $$ \log(x \#) > x \left(1 - \frac{1}{2 \log x} \right). $$ This is Theorem 4, formula 3.14.
The title of the article is Approximate Formulas for some Functions of Prime Numbers. It is pages 64-94, volume 6. I think I just found it online at some point...
YES, LINK
I did a simple experiment with C++, each line is just the low R-S bound, then a prime p, then log(p#), then the high R-S bound.
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
prime fac R-S low p log(p#) R-S high
1 0.557305 2 0.693147 3.4427
2 1.63464 3 1.79176 4.36536
3 3.44666 5 3.4012 6.55334
4 5.20136 7 5.34711 8.79864
5 8.70632 11 7.745 13.2937
6 10.4658 13 10.31 15.5342
7 13.9999 17 13.1432 20.0001
8 15.7736 19 16.0876 22.2264
9 19.3323 23 19.2231 26.6677
10 24.6939 29 22.5904 33.3061
11 26.4863 31 26.0244 35.5137
12 31.8766 37 29.6353 42.1234
13 35.4797 41 33.3489 46.5203
14 37.2837 43 37.1101 48.7163
15 40.8963 47 40.9602 53.1037
16 46.3254 53 44.9305 59.6746
17 51.7652 59 49.008 66.2348
18 53.5807 61 53.1189 68.4193
19 59.0327 67 57.3236 74.9673
20 62.6719 71 61.5863 79.3281
21 64.4928 73 65.8768 81.5072
22 69.96 79 70.2462 88.04
23 73.6084 83 74.665 92.3916
24 79.0861 89 79.1537 98.9139
25 86.3982 97 83.7284 107.602
26 90.0577 101 88.3435 111.942
27 91.8882 103 92.9782 114.112
28 95.5508 107 97.6511 118.449
29 97.3829 109 102.342 120.617
30 101.048 113 107.07 124.952
31 113.892 127 111.914 140.108
32 117.565 131 116.789 144.435
33 123.077 137 121.709 150.923
34 124.915 139 126.644 153.085
35 134.112 149 131.648 163.888
36 135.952 151 136.665 166.048
37 141.475 157 141.721 172.525
38 147 163 146.815 179
39 150.685 167 151.933 183.315
40 156.215 173 157.086 189.785
41 161.747 179 162.274 196.253
42 163.591 181 167.472 198.409
43 172.817 191 172.724 209.183
44 174.663 193 177.987 211.337
45 178.356 197 183.27 215.644
46 180.203 199 188.564 217.797
47 191.287 211 193.915 230.713
48 202.379 223 199.323 243.621
49 206.078 227 204.747 247.922
50 207.928 229 210.181 250.072
51 211.628 233 215.632 254.372
52 217.179 239 221.109 260.821
53 219.03 241 226.594 262.97
54 228.287 251 232.119 273.713
55 233.843 257 237.668 280.157
56 239.401 263 243.24 286.599
57 244.959 269 248.835 293.041
58 246.813 271 254.437 295.187
59 252.373 277 260.061 301.627
60 256.081 281 265.699 305.919
61 257.936 283 271.345 308.064
62 267.209 293 277.025 318.791
63 280.196 307 282.752 333.804
64 283.908 311 288.492 338.092
65 285.765 313 294.238 340.235
66 289.477 317 299.997 344.523
67 302.476 331 305.799 359.524
68 308.049 337 311.619 365.951
69 317.338 347 317.468 376.662
70 319.197 349 323.323 378.803
71 322.914 353 329.19 383.086
72 328.49 359 335.073 389.51
73 335.927 367 340.979 398.073
74 341.505 373 346.9 404.495
75 347.084 379 352.838 410.916
76 350.804 383 358.786 415.196
77 356.385 389 364.749 421.615
78 363.828 397 370.733 430.172
79 367.55 401 376.727 434.45
80 374.994 409 382.741 443.006
81 384.302 419 388.779 453.698
82 386.164 421 394.821 455.836
83 395.475 431 400.887 466.525
84 397.337 433 406.958 468.663
85 402.925 439 413.043 475.075
86 406.65 443 419.136 479.35
87 412.239 449 425.243 485.761
88 419.692 457 431.368 494.308
89 423.419 461 437.501 498.581
90 425.282 463 443.639 500.718
91 429.01 467 449.785 504.99
92 440.194 479 455.957 517.806
93 447.651 487 462.145 526.349
94 451.381 491 468.342 530.619
95 458.84 499 474.554 539.16
96 462.57 503 480.775 543.43
97 468.165 509 487.007 549.835
98 479.358 521 493.263 562.642
99 481.224 523 499.523 564.776
100 498.019 541 505.816 583.981
101 503.618 547 512.121 590.382
102 512.951 557 518.443 601.049
103 518.552 563 524.777 607.448
104 524.154 569 531.12 613.846
105 526.021 571 537.468 615.979
106 531.623 577 543.826 622.377
107 540.961 587 550.201 633.039
108 546.564 593 556.586 639.436
109 552.168 599 562.981 645.832
110 554.037 601 569.38 647.963
111 559.641 607 575.788 654.359
112 565.246 613 582.207 660.754
113 568.983 617 588.631 665.017
114 570.852 619 595.06 667.148
115 582.065 631 601.507 679.935
116 591.41 641 607.97 690.59
117 593.279 643 614.436 692.721
118 597.018 647 620.908 696.982
119 602.626 653 627.39 703.374
120 608.235 659 633.881 709.765
121 610.105 661 640.374 711.895
122 621.324 673 646.886 724.676
123 625.064 677 653.404 728.936
124 630.675 683 659.93 735.325
125 638.156 691 666.469 743.844
126 647.509 701 673.021 754.491
127 654.992 709 679.585 763.008
128 664.347 719 686.163 773.653
129 671.832 727 692.752 782.168
130 677.446 733 699.349 788.554
131 683.06 739 705.954 794.94
132 686.803 743 712.565 799.197
133 694.29 751 719.186 807.71
134 699.906 757 725.816 814.094
135 703.649 761 732.45 818.351
136 711.138 769 739.095 826.862
137 714.882 773 745.746 831.118
138 727.989 787 752.414 846.011
139 737.352 797 759.095 856.648
140 748.589 809 765.79 869.411
141 750.462 811 772.489 871.538
142 759.827 821 779.199 882.173
143 761.701 823 785.912 884.299
144 765.447 827 792.63 888.553
145 767.32 829 799.35 890.68
146 776.688 839 806.082 901.312
147 789.803 853 812.831 916.197
148 793.551 857 819.585 920.449
149 795.425 859 826.34 922.575
150 799.173 863 833.101 926.827
151 812.291 877 839.877 941.709
152 816.04 881 846.658 945.96
153 817.914 883 853.442 948.086
154 821.663 887 860.23 952.337
155 840.408 907 867.04 973.592
156 844.158 911 873.854 977.842
157 851.657 919 880.678 986.343
158 861.032 929 887.512 996.968
159 868.533 937 894.354 1005.47
160 872.283 941 901.201 1009.72
161 877.909 947 908.055 1016.09
162 883.535 953 914.914 1022.46
163 896.665 967 921.788 1037.34
164 900.416 971 928.667 1041.58
165 906.043 977 935.551 1047.96
166 911.671 983 942.442 1054.33
167 919.175 991 949.341 1062.82
168 924.803 997 956.245 1069.2
169 936.061 1009 963.162 1081.94
170 939.813 1013 970.083 1086.19
171 945.443 1019 977.009 1092.56
172 947.319 1021 983.938 1094.68
173 956.702 1031 990.876 1105.3
174 958.579 1033 997.816 1107.42
175 964.209 1039 1004.76 1113.79
176 973.593 1049 1011.72 1124.41
177 975.47 1051 1018.68 1126.53
178 984.855 1061 1025.64 1137.15
179 986.732 1063 1032.61 1139.27
180 992.363 1069 1039.59 1145.64
181 1009.26 1087 1046.58 1164.74
182 1013.01 1091 1053.57 1168.99
183 1014.89 1093 1060.57 1171.11
184 1018.65 1097 1067.57 1175.35
185 1024.28 1103 1074.57 1181.72
186 1029.91 1109 1081.59 1188.09
187 1037.42 1117 1088.6 1196.58
188 1043.06 1123 1095.63 1202.94
189 1048.69 1129 1102.66 1209.31
190 1069.35 1151 1109.71 1232.65
191 1071.23 1153 1116.76 1234.77
192 1080.62 1163 1123.81 1245.38
193 1088.13 1171 1130.88 1253.87
194 1097.53 1181 1137.95 1264.47
195 1103.16 1187 1145.03 1270.84
196 1108.8 1193 1152.12 1277.2
197 1116.31 1201 1159.21 1285.69
198 1127.59 1213 1166.31 1298.41
199 1131.35 1217 1173.41 1302.65
200 1136.98 1223 1180.52 1309.02
201 1142.62 1229 1187.64 1315.38
202 1144.5 1231 1194.75 1317.5
203 1150.14 1237 1201.87 1323.86
204 1161.41 1249 1209 1336.59
205 1170.81 1259 1216.14 1347.19
206 1187.73 1277 1223.29 1366.27
207 1189.61 1279 1230.45 1368.39
208 1193.37 1283 1237.6 1372.63
209 1199.01 1289 1244.77 1378.99
210 1200.89 1291 1251.93 1381.11
211 1206.53 1297 1259.1 1387.47
212 1210.29 1301 1266.27 1391.71
213 1212.17 1303 1273.44 1393.83
214 1215.93 1307 1280.61 1398.07
215 1227.21 1319 1287.8 1410.79
216 1229.09 1321 1294.99 1412.91
217 1234.73 1327 1302.18 1419.27
218 1266.7 1361 1309.39 1455.3
219 1272.34 1367 1316.61 1461.66
220 1277.98 1373 1323.84 1468.02
221 1285.5 1381 1331.07 1476.5
222 1302.43 1399 1338.31 1495.57
223 1311.84 1409 1345.56 1506.16
224 1325 1423 1352.82 1521
225 1328.77 1427 1360.09 1525.23
226 1330.65 1429 1367.35 1527.35
227 1334.41 1433 1374.62 1531.59
228 1340.05 1439 1381.89 1537.95
229 1347.58 1447 1389.17 1546.42
230 1351.34 1451 1396.45 1550.66
231 1353.23 1453 1403.73 1552.77
232 1358.87 1459 1411.01 1559.13
233 1370.16 1471 1418.31 1571.84
234 1379.57 1481 1425.61 1582.43
235 1381.45 1483 1432.91 1584.55
236 1385.21 1487 1440.21 1588.79
237 1387.1 1489 1447.52 1590.9
238 1390.86 1493 1454.83 1595.14
239 1396.51 1499 1462.14 1601.49
240 1407.8 1511 1469.46 1614.2
241 1419.09 1523 1476.79 1626.91
242 1426.62 1531 1484.12 1635.38
243 1437.91 1543 1491.47 1648.09
244 1443.56 1549 1498.81 1654.44
245 1447.32 1553 1506.16 1658.68
246 1452.97 1559 1513.51 1665.03
247 1460.5 1567 1520.87 1673.5
248 1464.27 1571 1528.23 1677.73
249 1471.8 1579 1535.59 1686.2
250 1475.56 1583 1542.96 1690.44
251 1488.74 1597 1550.33 1705.26
252 1492.51 1601 1557.71 1709.49
253 1498.16 1607 1565.1 1715.84
254 1500.04 1609 1572.48 1717.96
255 1503.8 1613 1579.86 1722.2
256 1509.45 1619 1587.25 1728.55
257 1511.34 1621 1594.64 1730.66
258 1516.99 1627 1602.04 1737.01
259 1526.4 1637 1609.44 1747.6
260 1545.23 1657 1616.85 1768.77
261 1550.88 1663 1624.27 1775.12
262 1554.65 1667 1631.69 1779.35
263 1556.53 1669 1639.11 1781.47
264 1579.14 1693 1646.54 1806.86
265 1582.9 1697 1653.98 1811.1
266 1584.79 1699 1661.42 1813.21
267 1594.2 1709 1668.86 1823.8
268 1605.51 1721 1676.31 1836.49
269 1607.39 1723 1683.76 1838.61
270 1616.81 1733 1691.22 1849.19
271 1624.35 1741 1698.68 1857.65
272 1630 1747 1706.15 1864
273 1635.65 1753 1713.62 1870.35
274 1641.3 1759 1721.09 1876.7
275 1658.26 1777 1728.57 1895.74
276 1663.91 1783 1736.06 1902.09
277 1667.68 1787 1743.55 1906.32
278 1669.56 1789 1751.04 1908.44
279 1680.87 1801 1758.53 1921.13
280 1690.29 1811 1766.03 1931.71
281 1701.6 1823 1773.54 1944.4
282 1709.14 1831 1781.05 1952.86
283 1724.22 1847 1788.58 1969.78
284 1737.41 1861 1796.1 1984.59
285 1743.06 1867 1803.64 1990.94
286 1746.83 1871 1811.17 1995.17
287 1748.72 1873 1818.71 1997.28
288 1752.49 1877 1826.24 2001.51
289 1754.37 1879 1833.78 2003.63
290 1763.8 1889 1841.33 2014.2
291 1775.11 1901 1848.88 2026.89
292 1780.76 1907 1856.43 2033.24
293 1786.42 1913 1863.99 2039.58
294 1803.39 1931 1871.55 2058.61
295 1805.27 1933 1879.12 2060.73
296 1820.35 1949 1886.69 2077.65
297 1822.24 1951 1894.27 2079.76
298 1842.98 1973 1901.86 2103.02
299 1848.64 1979 1909.45 2109.36
300 1856.18 1987 1917.04 2117.82
301 1861.84 1993 1924.64 2124.16
302 1865.61 1997 1932.24 2128.39
303 1867.49 1999 1939.84 2130.51
304 1871.27 2003 1947.44 2134.73
305 1878.81 2011 1955.05 2143.19
306 1884.47 2017 1962.66 2149.53
307 1893.9 2027 1970.27 2160.1
308 1895.78 2029 1977.89 2162.22
309 1905.21 2039 1985.51 2172.79
310 1918.41 2053 1993.13 2187.59
311 1927.84 2063 2000.77 2198.16
312 1933.5 2069 2008.4 2204.5
313 1944.82 2081 2016.04 2217.18
314 1946.71 2083 2023.68 2219.29
315 1950.48 2087 2031.33 2223.52
316 1952.36 2089 2038.97 2225.64
317 1961.8 2099 2046.62 2236.2
318 1973.11 2111 2054.28 2248.89
319 1975 2113 2061.93 2251
320 1990.09 2129 2069.59 2267.91
321 1991.98 2131 2077.26 2270.02
322 1997.64 2137 2084.93 2276.36
323 2001.41 2141 2092.6 2280.59
324 2003.3 2143 2100.26 2282.7
325 2012.73 2153 2107.94 2293.27
326 2020.28 2161 2115.62 2301.72
327 2037.26 2179 2123.3 2320.74
328 2059.9 2203 2131 2346.1
329 2063.68 2207 2138.7 2350.32
330 2069.34 2213 2146.4 2356.66
331 2076.89 2221 2154.11 2365.11
332 2091.98 2237 2161.82 2382.02
333 2093.87 2239 2169.54 2384.13
334 2097.64 2243 2177.25 2388.36
335 2105.19 2251 2184.97 2396.81
336 2120.29 2267 2192.7 2413.71
337 2122.18 2269 2200.42 2415.82
338 2125.95 2273 2208.15 2420.05
339 2133.5 2281 2215.89 2428.5
340 2139.17 2287 2223.62 2434.83
341 2144.83 2293 2231.36 2441.17
342 2148.6 2297 2239.1 2445.4
343 2159.93 2309 2246.84 2458.07
344 2161.82 2311 2254.59 2460.18
345 2182.58 2333 2262.34 2483.42
346 2188.24 2339 2270.1 2489.76
347 2190.13 2341 2277.86 2491.87
348 2195.79 2347 2285.62 2498.21
349 2199.57 2351 2293.38 2502.43
350 2205.23 2357 2301.15 2508.77
351 2218.45 2371 2308.92 2523.55
352 2224.11 2377 2316.69 2529.89
353 2227.89 2381 2324.47 2534.11
354 2229.77 2383 2332.24 2536.23
355 2235.44 2389 2340.02 2542.56
356 2239.21 2393 2347.8 2546.79
357 2244.88 2399 2355.58 2553.12
358 2256.21 2411 2363.37 2565.79
359 2261.87 2417 2371.16 2572.13
360 2267.54 2423 2378.96 2578.46
361 2280.75 2437 2386.75 2593.25
362 2284.53 2441 2394.55 2597.47
363 2290.19 2447 2402.36 2603.81
364 2301.52 2459 2410.16 2616.48
365 2309.08 2467 2417.97 2624.92
366 2314.74 2473 2425.79 2631.26
367 2318.52 2477 2433.6 2635.48
368 2343.07 2503 2441.43 2662.93
369 2360.07 2521 2449.26 2681.93
370 2369.51 2531 2457.1 2692.49
371 2377.06 2539 2464.94 2700.94
372 2380.84 2543 2472.78 2705.16
373 2386.51 2549 2480.62 2711.49
374 2388.4 2551 2488.47 2713.6
375 2394.06 2557 2496.31 2719.94
376 2414.84 2579 2504.17 2743.16
377 2426.17 2591 2512.03 2755.83
378 2428.06 2593 2519.89 2757.94
379 2443.17 2609 2527.75 2774.83
380 2450.73 2617 2535.62 2783.27
381 2454.51 2621 2543.5 2787.49
382 2465.84 2633 2551.37 2800.16
383 2479.07 2647 2559.25 2814.93
384 2488.51 2657 2567.14 2825.49
385 2490.4 2659 2575.02 2827.6
386 2494.18 2663 2582.91 2831.82
387 2501.74 2671 2590.8 2840.26
388 2507.41 2677 2598.69 2846.59
389 2513.08 2683 2606.59 2852.92
390 2516.85 2687 2614.48 2857.15
391 2518.74 2689 2622.38 2859.26
392 2522.52 2693 2630.28 2863.48
393 2528.19 2699 2638.18 2869.81
394 2535.75 2707 2646.08 2878.25
395 2539.53 2711 2653.99 2882.47
396 2541.42 2713 2661.89 2884.58
397 2547.09 2719 2669.8 2890.91
398 2556.53 2729 2677.71 2901.47
399 2558.42 2731 2685.63 2903.58
400 2567.87 2741 2693.54 2914.13
401 2575.43 2749 2701.46 2922.57
402 2579.21 2753 2709.38 2926.79
403 2592.44 2767 2717.31 2941.56
404 2601.89 2777 2725.24 2952.11
405 2613.23 2789 2733.17 2964.77
406 2615.11 2791 2741.1 2966.89
407 2620.78 2797 2749.04 2973.22
408 2624.56 2801 2756.98 2977.44
409 2626.45 2803 2764.92 2979.55
410 2641.57 2819 2772.86 2996.43
411 2654.8 2833 2780.81 3011.2
412 2658.58 2837 2788.76 3015.42
413 2664.25 2843 2796.71 3021.75
414 2671.81 2851 2804.67 3030.19
415 2677.48 2857 2812.63 3036.52
416 2681.26 2861 2820.58 3040.74
417 2698.28 2879 2828.55 3059.72
418 2705.84 2887 2836.52 3068.16
419 2715.29 2897 2844.49 3078.71
420 2720.96 2903 2852.46 3085.04
421 2726.63 2909 2860.44 3091.37
422 2734.19 2917 2868.42 3099.81
423 2743.64 2927 2876.4 3110.36
424 2754.99 2939 2884.38 3123.01
425 2768.22 2953 2892.37 3137.78
426 2772 2957 2900.37 3142
427 2777.67 2963 2908.36 3148.33
428 2783.34 2969 2916.36 3154.66
429 2785.23 2971 2924.35 3156.77
430 2811.7 2999 2932.36 3186.3
prime fac R-S low p log(p#) R-S high
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
A commonly studied arithmetic function is $$ \vartheta(x):=\sum_{p\leq x}\log(p)=\log(x\#). $$
The Prime Number Theorem can equally well be stated in terms of $\pi(x)$, $\psi(x)$, and $\vartheta(x)$. The equivalence of the statements is not difficult to show.
The classical error term for the Prime Number Theorem implies that there are constants $C_1,C_2>0$ such that $$ x\#<e^x e^{C_1x\exp\left(-C_2\sqrt{\log(x)}\right)}. $$ Basically any improvement on the error term in PNT improves this upper bound.