The Stieltjes constants _k are essentially the coefficients in the Laurent expansion of the Riemann zeta function about s=1. More specifically,

defines _k; the unsubscripted denotes Euler's constant, .577215... [A brief note on terminology: the _k are often called generalized Euler constants, and were studied by Stieltjes; hence sometimes they are instead known as Stieltjes constants. But occasionally the entire coefficient of (s-1)^k is called the kth Stieltjes constant. On this web page, we are interested in the values of the _k, and we'll refer to them as "Stieltjes constants", since for one thing it's faster to write than "generalized Euler constants".]

The first graphic displays the first two hundred Stieltjes constants (or rather generalized Euler constants...), in the form of the logarithm of their absolute value:

Similar figure, now for the first eight hundred Stieltjes constants; note the remarkable smoothness in the growth:

Next, we consider the ratio of the log of the absolute value of _k to k, first for all k from 1 to 800, then for k from 601 to 800; note the very regular "bumps":

The following figure indicates the ratio log(_k)/k for those _k's that are positive, for k=901 to 1000. This hopefully illustrates what we meant above by "very regular bumps":

Finally, we present some numerical values for _k.
[These were obtained by taking much higher precision values and having Mathematica display their values to 10, then 9, digits; I suspect these are therefore rounded to 10 digits for the first 100 constants I display, i.e. for ₁ to ₁₀₀ then rounded to 9 digits for the last 100, i.e. for ₇₀₁ to ₈₀₀. I am undecided as to how many digits I should bother displaying for each _k; values for _k to over 200 digits for k=1 to 300, over 150 digits for k=301 to 600, over 115 digits for k=601 to k=900, and over 80 digits for k=901 to 1000 are available on another web site, if someone is interested. (That site will soon be updated to include _k values through k=2000 to at least 110 digits, and up to k=2500 to at least 40 digits.) Here, I only supply high precision values for ₁₀₀ (to over 630 digits), ₈₀₀ (to over 350 digits), ₁₀₀₀ (to over 290 digits), ₁₄₀₀ (to over 200 digits), ₂₀₀₀ (to over 110 digits), ₂₂₀₀ (to over 110 digits), ₂₃₀₀ (to over 120 digits), and ₂₅₀₀ (to over 110 digits).]

₈₀₀ to over 350 digits: 4.9135405617183059420222476712064847339402719362442113026262039583988755867359\ 438843536710158276178530113623827558756099110315231608329143854160511826736713\ 972748825417058128811885077912051573099693648643298975867506236706481259739136\ 781051694396048167782007042868700364513020589867259952638325774220295015424602\ 54437793490578438931448149800385820097079043...*10³⁶⁹

₁₀₀₀ to over 290 digits: -1.5709538442047449345494023425120825242380299554570342998059351161258294099037\ 199854206254096008467812139555341596736867502331601668121071638479052640676685\ 423545768534756544207985919676028379296947261650838972395390475634805777368274\ 19020885874953467487775260240930149416316567894812928130663644516...*10⁴⁸⁶

₁₄₀₀ to over 200 digits: -4.0972873342343532394982319461419806083647517955771025010419784982373402100149\ 505890244128631999143411495412251669307998124239956025361257600688130640693133926385598135692293832350098923754453616825523628...*10⁷²⁸

₂₂₀₀ to over 20 digits: -1.222869680592907845550881...*10¹²⁴¹

The above four calculations each took less than three minutes; the calculation below took less than seven minutes. ₂₀₀₀ to over 110 digits: 2.680424678918000809504929834609356277409099264836410829548331346701687949728089727686622380790342607511360733839...*10¹¹⁰⁹

₂₂₀₀ to over 90 digits also took less than seven minutes, but nearly 8 minutes to get over 110 digits: -1.2228696805929078455508813376738703168396086351823059289770185794935419511856\ 18477984871036539175221112890314370286032444... *10¹²⁴¹

₂₃₀₀ to over 120 digits: 8.2283687746543004407015965275604433206784689826434521673412393372454407360957\ 6441279096701958066765438702305269666273908142400931... *10¹³⁰⁶

₂₅₀₀ to over 110 digits: 3.09222595656080208627358480085517110069857920869644722628396987113619118879848205835658417632121267290063723892135022544042321... *10¹⁴³⁹

The sign pattern for the first eight hundred constants is very orderly; here are their signs, in the form of a listing of the length of the "runs" of negatives and positives. More specifically, the listing below indicates that the first two values (i.e. ₁ and ₂) are negative, then the next three are positive, and so on, until the last eight computed in the listing above (i.e. ₇₉₃ to ₈₀₀ are seen to be positive):
2,3,4,3,4,5,4,5,5,5,5,5,6,6,5,6,6,6,6,6, 7,6,7,6,7,6,7,7,7,7,7,7,7,7,8,7,7,8,7,8, 7,8,7,8,8,7,8,8,8,8,8,8,8,8,8,8,8,8,8,9, 8,8,9,8,8,9,8,9,8,9,8,9,9,8,9,9,8,9,9,9, 8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,9,9, 9,10,9,9,9,8...

This page supplies a sample of some of the structure in the growth of the Stieltjes constants. Related structure appears in the growth of the generalized Stieltjes constants (i.e., up to normalization, the Laurent series coefficients for the Riemann-Hurwitz zeta function). Our method allows for computation of the Taylor coefficients for the zeta and the Riemann-Hurwitz zeta functions about other points besides s=1. Again, anyone interested in any of these computational matters can request a preprint using the email link at the top of the page.